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 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. Euler characteristic of covering space is product of degree of covering and Euler characteristic of base