  It is well known that any polycyclic group, and hence any finitely generated
nilpotent group, can be embedded into $GL_{n}(\mathbb{Z})$ for an appropriate
$n\in \mathbb{N}$; that is, each element in the group has a unique matrix
representation. An algorithm to determine this embedding was presented in
Nickel. In this paper, we determine the complexity of the crux of the algorithm
and the dimension of the matrices produced as well as provide a modification of
the algorithm presented by Nickel.
