We describe the connected components of the space of homomorphisms for a discrete nilpotent group. The connected components arising from homomorphisms with non-abelian image turn out to be homeomorphic to. We give explicit calculations when is a finitely generated free nilpotent group. In the second part of the paper, we study the filtration of the classifying space (introduced by Adem, Cohen and Torres-Giese), showing that for every, the inclusions induce a homology isomorphism with coefficients over a ring in which 2 is invertible. Most of the computations are done for and as well.