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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 aconjecture of Higman asserts that the number k(U(q)) of conjugacy classes inU(q) is given by a polynomial in q with integer coefficients. In an earlierpaper, the first and the third authors developed an algorithm to calculate thevalues of k(U(q)). By implementing it into a computer program using GAP, theywere able to calculate k(U(q)) for G of rank at most 5, thereby proving thatfor these cases k(U(q)) is given by a polynomial in q. In this paper we presentsome refinements and improvements of the algorithm that allow us to calculatethe values of k(U(q)) for finite Chevalley groups of rank six and seven, exceptE_7. We observe that k(U(q)) is a polynomial, so that the generalized Higmanconjecture holds for these groups. Moreover, if we write k(U(q)) as apolynomial 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 anexplicit formula for the coefficients of k(U(q)) of degrees zero, one and two.

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