Abstract
Let
$G(q)$
be a finite Chevalley group, where
$q$
is a power of a good prime
$p$
, and let
$U(q)$
be a Sylow
$p$
-subgroup of
$G(q)$
. Then a generalized version of a conjecture of Higman asserts that the number
$k(U(q))$
of conjugacy classes in
$U(q)$
is given by a polynomial in
$q$
with integer coefficients. In [S. M. Goodwin and G. Röhrle, J. Algebra 321 (2009) 3321–3334], the first and the third authors of the present paper developed an algorithm to calculate the values of
$k(U(q))$
. By implementing it into a computer program using
$\mathsf{GAP}$
, they were able to calculate
$k(U(q))$
for
$G$
of rank at most five, thereby proving that for these cases
$k(U(q))$
is given by a polynomial in
$q$
. In this paper we present some refinements and improvements of the algorithm that allow us to calculate the values of
$k(U(q))$
for finite Chevalley groups of rank six and seven, except
$E_7$
. We observe that
$k(U(q))$
is a polynomial, so that the generalized Higman conjecture holds for these groups. Moreover, if we write
$k(U(q))$
as a polynomial in
$q-1$
, then the coefficients are non-negative.
Under the assumption that
$k(U(q))$
is a polynomial in
$q-1$
, we also give an explicit formula for the coefficients of
$k(U(q))$
of degrees zero, one and two.