See two my comments above containing solution of the problem. I repeat: A nonabelian $p$-group $G$ has the center of index $p^2$ if and only if $G=S\text{Z}(G)$, where $S\le G$ is minimal nonabelian.

As this post contains a correct result, is it unclear why it is estimated by -4. This can do only unqualified person. Since only my posts estimated negatively, I decided to delete all them.

Se two my comments above containing solution of the problem. I repeat: A nonabelian $p$-group $G$ has the center of index $p^2$ if and only if $G=S\text{Z}(G)$, where $S\le G$ is minimal nonabelian.

See two my comments above containing solution of the problem. I repeat: A nonabelian $p$-group $G$ has the center of index $p^2$ if and only if $G=S\text{Z}(G)$, where $S\le G$ is minimal nonabelian.