Cohomology of finite groups with finite coefficients I'm wondering if the group cohomology of a finite group $G$ can be made nontrivial with a nice choice of a finite $G$-module M. In other words, given a finite group $G$ and a number $n$, does there exist a finite $G$-module $M$ such that $H^n(G,M)$ is non-zero?
I would also be interested in the special case that $G$ is a finite $p$-group and n = 2. Can I always get $H^2(G,M) \ne 0$ for some finite $M$?
Thanks for your help.

It is well known that, If $G$ and $M$ are finite $p$-groups, $H^2(G,M)=0$ is equivalent to $H^i(G,M)=0$ for all $i \geq 0$. 
If one take any $M$ on which $G$ acts trivially, then $H^1(G,M) \cong Hom(G,M)$, which is not trivial unless $M$ or $G$ are trivial.
