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 $BG$, $H_{top}^d[BG, M]=H^d[G, M]$.)
For the latter three, here is the integercoefficients (apply Kunneth formula to get your mod2 coefficients: 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). 

