  We study the Lie ring $\mathfrak{nil}_n$ of all strictly upper-triangular
$n\!\times\!n$ matrices with entries in $\mathbb{Z}$. Its complete homology for
$n\!\leq\!8$ is computed. We prove that every $p^m$-torsion appears in
$H_\ast(\mathfrak{nil}_n;\mathbb{Z})$ for $p^m\!\leq\!n\!-\!2$. For $m\!=\!1$,
Dwyer proved that the bound is sharp, i.e. there is no $p$-torsion in
$H_\ast(\mathfrak{nil}_n;\mathbb{Z})$ when prime $p\!>\!n\!-\!2$. In general,
for $m\!>\!1$ the bound is not sharp, as we show that there is $8$-torsion in
$H_\ast(\mathfrak{nil}_8;\mathbb{Z})$. As a sideproduct, we derive the known
result, that the ranks of the free part of
$H_\ast(\mathfrak{nil}_n;\mathbb{Z})$ are the Mahonian numbers (=number of
permutations of $[n]$ with $k$ inversions), using a different approach than
Kostant. Furthermore, we determine the algebra structure (cup products) of
$H^\ast(\mathfrak{nil}_n;\mathbb{Q})$.

Algebraic Topology