  In this paper, we classify the congruence classes of
$\mathbf{F}^5_{n(2)}$-polyhedra, i.e. $(n-1)$-connected, at most
$(n+5)$-dimensional polyhedra with 2-torsion free homology. The proof relies on
the matrix problem technique which was developed in the classification of
representations of algebras and applied to homotopy theory by Baues and Drozd.
