Let $G$ be a compact, connected, simply-connected Lie group with centre $Z(G)$, and consider the Lie group $G/Z(G)$. I believe that for $G$ a classical group, the Lie group $G/Z(G)$ is sometimes called a projective classical group. What is known about the integral cohomology $H^*(G/Z(G);\mathbb{Z})$? I am particularly interested in the integral cohomology of the projective special unitary group $PSU(n)$. I would appreciate any and all references/suggestions.

Thanks!