While working through a prior problem on classifying a group of order $625$, I stumbled upon (by virtue of a mistaken answer by a user), the following problem. Since the dimension of irreducible representations divide the order of a group and have their sum of squares equal to the order of the group, how can we determine the number of conjugacy classes of the $625$ order group with center of order $25$?
Specifically, the center of the group would correspond to $25$ trivial representations of dimension $1$. Now, how to show that there are $120$ other irreducible representations using only representation/ character properties, as we must have $625=1^2A+5^2B+25^2C+125^2D+625^2E$, ($A,B,C,D,E$ are number of irreducible representations of degrees $1, 5, 25, 125, 625$ respectively) which does not give us anything special? Any hints? Thanks beforehand.