Let $G$ be the group generated by $a\_i,b\_j$, $i,j=1,2,3$ and the following relations:
$$b\_ib\_j=b\_jb\_i, a\_ib\_j=b\_ja\_i, a\_1a\_2=b\_3a\_2a\_1, a\_2a\_3=v\_1a\_3a\_2, a\_3a\_1=v\_2a\_1a\_3$$
I want to comute the homology of $H\_nG=H\_n(G,\mathbb{Z})$ where $\mathbb{Z}$ is considered as a trivial $G$-module.
I have made some attempts of applying spectral sequences (Lyndon-Hochschild-Serre). Here I chose $N=\langle b\_1,b\_2,b\_3\rangle$ as the normal (central) subgroup of $G$. However I could not manage to compute the differentials. Does anyone know how to do this (either by spectral sequences or any other way)?



Homology of a certain group
If $v\_i=b\_i$, your group is the free nilpotent group of class 2 and rank 3. Its homology is known: <http://iopscience.iop.org/1064-5616/189/4/A03>. 


