Suppose $G$ is a classical matrix group over a finite field of order $q$.

If $C$ is a conjugacy class in $G$ , is $|C|$ a polynomial in $q$?

This question is supported by the fact that whenever I have calculated all conjugacy classes and their sizes for very small groups (for example, $GL_{2}(\mathbb{F}_{q})$, $GL_{3}(\mathbb{F}_{q})$, $SL_{2}(\mathbb{F}_{q})$, $SL_{3}(\mathbb{F}_{q})$) the sizes of conjugacy classes turn out to be polynomial in $q$.

So, does this property hold for all finite classical group, or at least the case of $GL_n$ or $SL_n$?

aconjugacy class for different $q$'s. Even the number of conjugacy classes of $PSL(2,q)$ dependes on the parity of q. $\endgroup$ – LeechLattice Apr 18 '19 at 1:34familyof conjugacy classes over afamilyof groups. You have to be careful how you choose these families for this to make sense. $\endgroup$ – user44191 Apr 18 '19 at 1:35