Let \(\mathcal L(r)\) be a free Lie algebra over \(\mathbb C\) of rank \(r\), \(H_i(r)\) the vector space of homogeneous Lie polynomials of degree \(i\) in \(\mathcal L(r)\). The free \(N\)-step nilpotent Lie algebra of rank \(r\) is the Lie algebra \[ \mathcal L(N, r)=\frac{\mathcal L(r)}{\sum_{i\geq N+1}H_i(r)}. \] In the paper some general results on homology of \(\mathfrak n=\mathcal L(N, r)\) as a \(GL(r,\mathbb C)\)-module are obtained: the relation between the structure of \(GL(r,\mathbb C)\)-modules \(H_{n-i}(\mathfrak n)\) and \(H_i(\mathfrak n)\) (Poincaré duality); a stabilization of Young diagrams of \(H_i(\mathfrak n)\) with respect to \(r\); boundaries for minimal weights of \(H_i(\mathfrak n)\). The results of the calculations of the Young diagrams are given for all \(i\) in the cases \(\mathfrak n=\mathcal L(III,2),~\mathcal L(IV,2),~\mathcal L(V,2), ~\mathcal L(III, 3)\) and for \(i=1,\ldots , 4\) for all \(r\) in the case \(\mathfrak n=\mathcal L(III,r)\).