  Let $q$ be a power of a prime $p$. Let $P$ be a parabolic subgroup of the
general linear group $\GL_n(q)$ that is the stabilizer of a flag in $\FF_q^n$
of length at most 5, and let $U = O_p(P)$. In this note we prove that, as a
function of $q$, the number $k(U)$ of conjugacy classes of $U$ is a polynomial
in $q$ with integer coefficients.
