On affine classification of permutations on the space GF(2)3

We give an elementary proof that by multiplication on left and right by affine permutations A, B is an element of AGL(3, 2) each permutation pi:GF(2)(3)-> GF(2)(3) may be reduced to one of the 4 permutations for which the 3x3-matrices consisting of the coefficients of quadratic terms of coordinate functions have as an invariant the rank, which is either 3, or 2, or 1, or 0, respectively. For comparison, we evaluate the number of classes of affine equivalence by the Polya enumerative theory.