Calculating conjugacy classes in Sylow p-subgroups of finite Chevalley groups of rank six and seven

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. Rohrle, J. Algebra 321 (2009) 3321-334], 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 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.