  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 an earlier
paper, the first and the third authors developed an algorithm to calculate the
values of k(U(q)). By implementing it into a computer program using GAP, they
were able to calculate k(U(q)) for G of rank at most 5, 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.
