If a group $G$ is defined by a single relation $R$, where $R$ = $Q^q$ for $q$ maximal, and if $K$ is any left $G$-module, then
$H^2(G,K) \cong K/(\frac{\partial R}{\partial x_1},\dots,\frac{\partial R}{\partial x_m})$ and $H^n(G,K) \cong 0, \ \ n>2$
This is Corollory 11.2, in a paper from R.C. Lyndon.
What does maximal mean here, and, is there anything known about $H^1(G,K)$ and $H^0(G,K)$? Here I think for instance of a group $G$ having a presentation of the form $\langle x_1\lvert x_1^2\rangle $.
Thank you for your assistance and help.