At a rally in Dresden on Monday, which German media said drew about 8,000 people, Lutz Bachmann said Social Democrat (SPD) minister Heiko Maas was the "worst spiritual fire raiser" since Goebbels and Karl-Eduard von Schnitzler.
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Von Schnitzler was a television commentator in Communist East Germany who strongly criticised Western governments and media.
The comment is the latest in a series of provocative remarks made at the regular rallies of Patriotic Europeans Against the Islamisation of the West (PEGIDA). Only two weeks ago, a speaker said that concentration camps were "unfortunately out of action".
The refugee crisis in Europe has boosted the popularity of PEGIDA's rallies in the eastern city of Dresden and raised fears about right-wing radicalism. Many voters are worried about how Germany will cope with an influx of about 1 million migrants this year, many fleeing wars in the Middle East and Africa.
Social Democrats, who share power with Chancellor Angela Merkel's conservatives, expressed outrage at Bachmann's comment. SPD General Secretary Yasmin Fahimi said it was perfidious and disgusting.
A spokeswoman for Dresden prosecutors said they had started an investigation into slander.
Bachmann said on his Facebook page that he would not be silenced. "If the Sharia Party of Germany (SPD) and the whole press ... demand hundreds of thousands of investigations, YOU WILL NOT GAG ME! I will still say openly say what I think."
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Bachmann has already been charged by Dresden prosecutors with incitement because of a post on social media last year in which he described refugees and asylum seekers as "animals" and "scumbags". No trial date has yet been set.
He quit as leader of PEGIDA earlier this year after a photo was published of him posing as Hitler which led to internal squabbles and the grassroots movement all but fizzled out, before the migrants crisis swept Europe.
Support for Merkel's conservatives has dropped over her handling of the refugee crisis while the anti-immigrant Alternative for Germany (AfD) is up in opinion polls.
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