On the homology of a free nilpotent group of class 2

Let $G$ be a free nilpotent group of class 2, and let ${\mathscr G}$ be a free nilpotent Lie ring of class 2 with the same number of free generators. For $G$ a free resolution is constructed which as a graded ${\mathbb Z}G$-module is isomorphic to ${\mathbb Z}G\otimes \Lambda ({\mathscr G})$, where ${\mathbb Z}G$ is the group ring of the group $G$ and $\Lambda ({\mathscr G})$ is the exterior algebra of the ring ${\mathscr G}$. As a consequence of the basic construction an isomorphism $H_nG\cong H_n{\mathscr G}$ of integral homology is derived.