Let G n,p be the Sylow p-subgroup of SL( n, p) formed by the upper unitriangular matrices. The aim of this paper is to describe algorithms for the computation of the number of conjugacy classes, the conjugacy vector of G n,p , the character (rational or real) of the elements of G n,p , the cardinality of the centralizer of each matrix of G n,p, the conjugacy vector of the normal subset N π corresponding to a pivot disposition π, and the character (inert or ramification) of each entry of any matrix of G n,p . For p=2, by using these algorithms, we have proved that Kirillov's conjecture, every matrix of G n,2 is conjugate to its inverse, holds for n⩽12, but for n=13 there exists a unique pair of inverse conjugacy classes not conjugate. A representative pair of these conjugacy classes is given in [J. Algebra 202 (1998) 704]. For n=14, we give the complete list of the canonical matrices of the 22 counterexamples to Kirillov's conjecture. For n⩽14, we have proved that A and A 5 are conjugate and for n=25 we have found a matrix A∈ G 25,2 such that A and A 5 are not conjugate. In addition, for n=32 we have found a matrix A∈ G 32,2 such that A and A −1 are conjugate but A and A 5 are not conjugate. So, Isaacs' conjecture, every real matrix in G n,2 m is actually rational, is not true. For p=3, Isaacs and Kereguezian give in [ibid.] a matrix A∈ G 20,3 such that A and A 4 are not conjugate. In this paper, for every odd prime p, we obtain a matrix A∈ G n,p , with n=6 p+1 such that A and A 1+ p are not conjugate. Consequently, there exist irreducible characters of G 6p+1,p that are not Q(ϵ p) -valued, where ϵ p is a primitive pth root of unity. Besides, for p=3 and n⩽13 we have computed the conjugacy vector of G n,3 and verified that every matrix A∈ G n,3 is conjugate to A 4. Thus, for all n⩽13, the character values of G n,3 always lie in Q(ϵ 3) .