Consider the following situation: I have given a finite $p$-group $P$ (in the case I am interested in $p = 2$) with cyclic center $Z(P)$ and I also know the structure of the quotient $P/Z(P)$ (which is non-trivial). What are the possible isomorphism types of $P$? Of course, in general a central extension such as $P$ is not unique. However, I wonder whether in this particular situation more can be said about the structure of $P$. For example, $P$ cannot be the trivial extension, because otherwise the center would be too large.