* Greens set for best showing yet in EU election
* Role seen in "progressive coalition"
* But Greens will still be one of the smaller groups (Updates with more details)
By Alissa de Carbonnel
BRUSSELS, May 26 (Reuters) - Surfing a global wave of climate activism to their strongest showing yet in European Union elections, Europe's Greens on Sunday readied to press their demands on climate and trade in Brussels.
An ebb in support for mainstream parties raised hopes among Europe's ecology parties that they could use their still relatively small presence in the EU assembly to act as kingmakers in an increasingly fragmented EU legislature.
Many of the gains came from the northern European countries that were once the continent's industrial heartlands where, often inspired by 16-year-old Swedish climate activist Greta Thunberg, young people have taken to the streets to demand a break from a legacy of dependence on fossil fuels.
Germany's Greens leapt into second place behind Chancellor Angela Merkel's CDU conservatives, with a third of its voters under the age of 30.
"This election was above all about the issue of climate and climate protection," acknowledged Annegret Kramp-Karrenbauer, leader of the German Christian Democrats (CDU), currently the biggest European party in the EU chamber.
Greens also doubled their share of the vote to take two seats in Finland and Denmark, won their first two seats in Ireland in two decades, grabbed third place in France and were set for strong showings in Belgium, the Netherlands and Britain - the latter partly as a result of being clearly anti-Brexit.
The Greens added at least 15 more seats in Parliament, where there will be now weeks of bargaining among all the parties to form a stable majority. The stakes are high amid fears by some that unprecedented gains by populist nationalists will seek to hijack or block a pro-European agenda.
Leaders of the pan-European Green alliance projected said their support will not come cheap.
Not only will they seek written commitments on climate action - which could mean tighter regulation for industry - but will push demands on tax policy and in trade negotiations with Australia and the United States.
The Greens also talk tough on using the threat of cutting off access to EU funds over violation of the bloc's rule of law principals in member states like Hungary and Poland - an issue where they will find common cause with the liberals.
"The citizens are giving a bigger lever than we ever had in this parliament and we are going to use that lever," Philippe Lamberts, who leads the Greens family of more than 30 national parties in Parliament, told Reuters.
The 751-member house has emerged as a more ambitious voice on a raft of regulation to slash emissions in the world's biggest economy over its last five-year term than national capitals or the EU executive.
It faced down Germany and its powerful automotive lobby to push for deeper cuts to car pollution, higher targets on renewable use and energy savings, limits on plastic waste and palm oil over deforestation. It pushed for more transparency in free trade deals and close tax loopholes for corporations.
A festive mood reigned at the Greens headquarters in the European Parliament on Sunday night at the end of campaign, which profited from worries over global warming but also frustration at stagnating living standards and disillusionment with establishment parties.
Still ecologists made little inroads - and even lost some seats - in eastern and southern Europe, where climate worries are dwarfed by concerns over migration and jobs.
And projected gains in Britain will be short-lived if the country leaves the EU as planned. As the parliament shrinks after Brexit, 27 of Britain's 73 seats will be redistributed to candidates elected onto a reserve list in other countries.
There the Green group stands to lose - although they expect to be able to add some seats by courting unaffiliated parties like the Pirate party in the Czech Party.
The centre-right and centre-left parties could yet turn to the liberal alliance of ALDE and French president Emmanuel Macron's party to maintain their majority in informal coordination on voting over the last five years.
But they have all three leading parties have courted the Greens in an effort to bolster their credential on climate - one of the areas where the EU has the most say over policy.
"We know they want to court us," Lamberts said. "It won't be that you get the Green votes for forever and that we will forget about the promises."
(Reporting by Alissa de Carbonnel in Brussels Additional reporting by Andreas Rinke in Berlin, Francesco Guarascio and Bart Biesemans in Brussels Editing by Mark John)
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