Communism as an ideology and practise came first, but not by far. The USSR was official created in 1924, while Fascism's first, Mussolini came to power in 1925. Hitler got power in 1933 with is own extreme sort of Fascism, what is now known as Nazism.
On one and only agreement between Communism and Nazism is Totalitarianism. The total and utter rule of the people, controlling every aspect of their lives. Other than that, they are polar opposite.