I would like to know what are the group cohomology classes $H^d[Z\_n, Z\_2]$, $H^d[U(1), Z\_2]$, $H^d[SO(n), Z\_2]$, $H^d[SU(n), Z\_2]$, etc. Thanks!
(Here the group cohomology $H^d[G, M]$ for a group $G$ is the topological cohomology of the classifying space $BG$, $H\_{top}^d[BG, M]=H^d[G, M]$.)



Group cohomology with $Z_2$ coefficient
For the latter three, here is the integer-coefficients (apply Kunneth formula to get your mod-2 coefficients:


[Group cohomology of compact Lie group with integer coeffient](https://mathoverflow.net/questions/75389/group-cohomology-of-compact-lie-group-with-integer-coeffient) 


As for the first: $H^i(\mathbb{Z}\_n)$ is $\mathbb{Z}\_n$ for $i$ even, and zero otherwise. Again, apply Kunneth formula for $\mathbb{Z}\_2$-coefficients.


In general, for a finite group $G$, $H^\*(G,M)$ is a $\mathbb{Z}\_{|G|}$-module for $n>0$. And If $M$ has exponent $p$ (prime) then $H^\*(G,M)$ is a $\mathbb{Z}\_p$-vector space. So for $gcd(n,2)=1$ we must have $H^\*(\mathbb{Z}\_n,\mathbb{Z}\_2)=0$ (in positive dimensions).


