Addressing a trial which would have been farcical if it had not led to grim penal colonies, Pussy Riot member Maria Alekhina asked:

"How did our performance, a small and somewhat absurd act to begin with, balloon into a full-fledged catastrophe?" Masha Gessen's impassioned first draft of the group's history attempts to explain.

In 2011, a group of young artists planned an "action" to highlight a minor absurdity of the Russian state. The notoriously violent "militia" had been rebranded the "police", as if that would change their reputation - a bit of Newspeak familiar from Russian history and not unknown here either. So the art collective Voina ("War") decided to take the new, friendlier image at face value by asking for directions and then kissing the cops with fervent gratitude, filming the results.

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But the group had decided to make the kisses same-sex, and the firmly heterosexual men of Voina couldn't quite face it, backing off with feeble excuses. In the end, "Buss The Buzz" (as Gessen translates it) became an all-female action, led by the fiercely political Nadezhda Tolokonnikova, known as Nadya. She declared: "Madame Officer, have you heard your boss, the mayor's, speech in which he said that 'Moscow does not need gays'? No? That's reason enough for me to suck your face."

Voina had been trying to find a new way of making provocative art in a post-glasnost society where censorship was creeping back. They had had sex in the Biology Museum in Moscow; re-staged the hanging of 19th-century revolutionaries in a supermarket; drawn a penis on a drawbridge to make it look like it was entering the secret police headquarters; mocked traffic police taking bribes by dressing up as their nagging families; and staged other actions which combined juvenile silliness with serious political satire.

But after "Buss The Buzz", Nadya, 24, and her ally, Yekaterina Samutsevich, 30, known as Kat - a computer engineer turned photographer - were moving away from the male-run, Art-with-a-capital-A strategy of Voina.

Shortly afterwards, they formed Pisya Riot - "pisya" being a toddler's word like wee-wee or pee-pee. Not coincidentally, Nadya was the mother of a toddler herself. So was Maria, 25, an environmental campaigner who joined the renamed Pussy Riot, along with a few others.

They were never really a band. They borrowed tunes from old British Oi! groups like Cockney Rejects and Angelic Upstarts. Wearing dresses and colourful balaclavas, they screamed or declaimed their lyrics, rather than sang. They didn't play gigs, they made video clips of performances, where the settings - the Metro, designer boutiques, a fashion show, a detention centre - were carefully targeted to be part of the message.

And what was that message? The details varied, but in an authoritarian state, with traditional forms of protest either crushed or tainted by historical precedent, just being their angry, mischievous selves was a challenge. At Red Square in January 2012, they climbed on a high stone platform and chanted: "A Russian riot means we exist."

In hindsight, their next action, at the Cathedral of Christ the Saviour, was doomed. They justified their target by pointing to the Orthodox Patriarch's staunch support for Russian president Vladimir Putin, as well as the cathedral's embrace of consumerism via an in-house luxury car wash and banquet hall.

But one Pussy Riot member left the country beforehand after, bizarrely, a warning from her astrologer; another chickened out on the morning of February 21. The remaining five - two remain anonymous - arrived at the cathedral to find news had leaked to too many journalists. There was no service being held, as has sometimes been reported, nor did they dance around the altar. Kat was carried out by a security guard just as she unpacked a guitar; the others rushed through a 40-second "performance" and then had to rush out a half-baked video, annoyed that they'd screwed up.

For that - and despite support then and since from Western pop stars - Nadya and Maria were sentenced to two years in penal colonies (Kat's sentence was eventually suspended, after she sacked their lawyers, who were clearly out of their depth).

And, as Gessen's account makes brutally clear, it really was hard time: 12- to 17-hour work shifts, with humiliating restrictions on basic hygiene. They had little contact with the outside world, and were hundreds of miles from home. Maria threw herself into legal challenges to the conditions; both went on hunger strike. They were released before Christmas, two months early, unrepentant but surely scarred.

Gessen's book is vividly told. She doesn't pretend to be neutral, having previously written a scathing biography of Putin and organised direct action herself. But this doesn't stop her being critical of some of Pussy Riot's speeches and actions, while making her views clear about their lawyers and Nadya's demanding father. Yet overall this is an angry book, which makes it shockingly clear what an injustice was done in a ludicrous trial - and what the treatment of these women says about the system in Putin's Russia.