AGL(m, 2) Acting on R(r, m)/R(s, m)

Let AGL(m, F) be the general affine group of order m over an arbitrary field F. We determine the conjugacy classes of AGL(m, F). When F = GF(q), we also determine the sizes of the centralizers of the elements in AGL(m, F). The group AGL(m, 2) (= AGL(m, GF(2))) acts on each of the Reed-Muller codes of length 2m as an automorphism group. We denote the rth order Reed-Muller code of length 2m by R(r, m) and prove that under the action of AGL(m, 2) the number of orbits of R(t, m)/R(s, m) is equal to that of R(m − (s + 1), m)/R(m − (t + 1), m) for −1 ≤ s < t ≤ m. We also compute the numbers of AGL(m, 2)-orbits of R(t, m)/R(s, m) for m = 6, 7, and −1 ≤ s < t ≤ m.