Calculating conjugacy classes in Sylow p-subgroups of finite Chevalley groups

In [S.M. Goodwin, On the conjugacy classes in maximal unipotent subgroups of simple algebraic groups, Transform. Groups 11 (1) (2006) 51–76, §8], the first author outlined an algorithm for calculating a parametrization of the conjugacy classes in a Sylow p-subgroup U(q) of a finite Chevalley group G(q), valid when q is a power of a good prime for G(q). In this paper we develop this algorithm and discuss an implementation in the computer algebra language GAP. Using the resulting computer program we are able to calculate the parametrization of the conjugacy classes in U(q), when G(q) is of rank at most 6. In these cases, we observe that the number of conjugacy classes of U(q) is given by a polynomial in q with integer coefficients.