We have the following two facts about group cohomology: \begin{eqnarray} H^2(\mathbb{Z}_n\times\mathbb{Z}_n,U(1))=\mathbb{Z}_n;\\ H^3(\mathbb{Z}_n,U(1))=\mathbb{Z}_n. \end{eqnarray} My question is whether there exists the following generalization:
"There exists a monomorphism from $H^{d+1}(G,U(1))$ to $H^2([C(G)]^d,U(1))$ where $C(G)$ is the center of $G$ (or maybe we can first restrict to abelian $G$ or finite $G$)"?