Group with periodic cohomology

Definition
A group $$G$$ is said to have periodic cohomology if there exists a cohomology class $$c \in H^d(G;\mathbb{Z})$$ for some positive integer $$d$$, such that the cup product with $$c$$ defines an isomorphism between $$H^m(G;\mathbb{Z})$$ and $$H^{m+d}(G;\mathbb{Z})$$ for every $$m$$. In particular, the sequence of cohomology groups is periodic.

A finite group with periodic cohomology is a finite group that has periodic cohomology.