Classifying space of nontrivial finite group cannot have finitely generated homology

Statement
Suppose $$G$$ is a nontrivial finite group, viewed as a discrete group. Then, the classifying space of $$G$$ (in fact, any classifying space, the precise choice does not matter since they are all homotopy-equivalent spaces) is not a fact about::space with finitely generated homology.

Facts used

 * 1) A contractible space, and more generally, a weakly contractible space, is a space with Euler characteristic one, i.e., its Euler characteristic is one.
 * 2) uses::Euler characteristic of covering space is product of degree of covering and Euler characteristic of base