It was 27 years ago when the Jim Sillars, flush from winning the Govan by-election for the SNP, taunted Scottish Labour MPs for being the "feeble 50", unable to defend Scotland against the excesses of Thatcherism. As the Westminster Parliament begins its summer recess, we can reflect on what, if anything, the 56 SNP MPs have been able to achieve.

Parliamentary arithmetic militates against SNP influence. They may have 56 out of 59 MPs in Scotland but in the Commons they are just 56 out of 650. The SNP total could have mattered more in the context of a hung parliament, but majority government means that the Conservatives will win the votes so long as the parliamentary party remains united. The adversarial and non-territorial structure of the Commons has also created the faintly ridiculous situation where the two most prominent Scottish posts available to MPs – the Secretary of State for Scotland and the shadow secretary of state for Scotland – are held by parties that each secured a single Scottish Member of Parliament.

But the SNP does have a bigger voice and presence that has not gone unnoticed among the Westminster establishment. As the third-biggest party, the SNP is granted more status, more rights and more parliamentary time. Where it has found common cause with the other opposition parties, the SNP can claim at least some success in influencing a few UK government decisions, for example, confirming that the EU referendum will not coincide with the elections to the devolved parliaments and the decision to postpone the introduction of English votes on English Laws. Most controversially, the SNP signalled its intent to vote on fox-hunting legislation should it come before parliament, forcing the Government to defer the legislation. But the latter was clearly a response, payback perhaps, to the lack of persuasive influence SNP MPs had been able to exert in the committee stages of the Scotland Bill, where every SNP amendment was rejected.

The SNP will also bring a more explicitly territorial dimension to parliamentary committees, where much of the important work of the Commons is carried out. As well as holding the chair of the Scottish Affairs committee and the Energy and Climate Change committee, 29 SNP members are represented on 26 parliamentary committees, including the select committees on mainly reserved matters – Foreign Affairs, Defence, Work and Pensions, Home Affairs and so on – as well as those whose main business is devolved to the Scottish Parliament: Health, Education and Communities and Local Government. Scottish MPs have, of course, been represented on some of these committees in previous parliaments. The difference is that the SNP members bring both a Scottish perspective and a nationalist perspective across Westminster parliamentary politics.

But the biggest test of influence may be behind the scenes. SNP success in the Westminster elections was always going to be used as added leverage for the SNP Government in its relationship with the UK Government. The SNP Government can legitimately claim to speak for Scotland in a way the UK Government cannot. Scottish and UK government ministers have been meeting regularly since the election to discuss the new devolution proposals and the new fiscal framework. The Secretary of State for Scotland will bring new amendments to the Scotland Bill to the House of Commons after summer. It is not clear how extensive those amendments will be, nor whether they will meet some of the Scottish Government’s demands. Nor is it clear if the SNP’s reminder, after their decision to vote on fox hunting in England, to "an arrogant UK government of just how slender their majority is" will heighten the Scottish Government’s influence or stiffen the UK Government’s resolve.

The capacity to exert influence over the Scotland Bill may be the SNP’s biggest test. Even if it doesn’t win votes in parliament, it can win the debate and increase the likelihood for a further round of devolution reforms. But the devolution legislation is not just a test for the SNP. It is also a test for the UK Government and, by extension, for the Union. The SNP sent 56 MPs to the Commons because Scots voted for them in unprecedented numbers. If the Westminster Parliament can accommodate these voices, it can heighten its status and relevance for Scottish politics. On the other hand, ignoring the democratic legitimacy the SNP MPs carried with them from Scotland would be to play a very high stakes game with the future of the Union.

Nicola McEwen is Professor of Territorial Politics at the University of Edinburgh and Associate Director of the Centre on Constitutional Change (@CCC_Research)