This is a problem from Preliminary Exam - Spring 1984, UC Berkeley
For a $p$-group of order $p^4 $, assume the center of $G$ has order $p^2 $. Determine the number of conjugacy classes of $G$.
What I have tried: each element of the center constitutes a conjugacy class; the other conjugacy classes have order a power of $p$; their sum is $ \ p^{4} - p^{2}$.