the group of the upper unitriangular matrices over F, , the finite field with q -p' elements, and we determine the number of classes of G n modulo ( g -1 ) 5 . In this paper we maintain the notation given in [1, 2] . We count the canonical matrices by firstly fixing the configurations of entries with non-zero values which correspond to them. LEMMA 1 . All the matrices in Q n having exactly one non-zero value oSthe main diagonal are canonical. The number of them is (ii • (q -1), wAere fii -n(n

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... )/2. PROOF: Let A = I n + aijEij, with a,j ^ 0. We must prove that A has the value zero at all of its inert points. The only entry with a non-zero value is (i,j) and we have Lij = a^j+jZi+ij + • • • + ai t j-\Xi-\ t j -(oi+ijXj t i+i + • • • + Oj_ijijj_i) = 0. Hence (i,j) is a ramification point. Finally, all these matrices are canonical. The number of them is deduced taking into account that we can have |F q -{0}| = q-1 different values at the non-zero entry and that the total number of such entries that we can consider is /xj = n ( n -l ) / 2 . D LEMMA 2 . A matrix A of Q n which has exactly two non-zero values off the main diagonal is canonical if and only if these values are situated in entries which are not in the same row or column. The number of such matrices is fi2 • (q -I) 2 , wAere H 2 = n{n -l)(n -2)(3n -5)/24. PROOF: Let A = I n + a^Eij + a T .E T " with a^ ^ 0 ^ a T ,. We show that A is a canonical matrix if and only if the entries (r, a) and (i,j) are not in the same row or column. If that condition is satisfied, (i,j) and (r,s) are ramification points, since