A study of the Wamsley group and its Sylow subgroups

We study the Wamsley group $\langle X,Y,Z\,|\, X^Z=X^\alpha, {}^Z Y=Y^\beta, Z^\gamma=[X,Y]\rangle$ and its Sylow subgroups, where $\alpha^\gamma\neq 1\neq \beta^\gamma$ and $\gamma>0$, obtaining the sharpest results when $\alpha=\beta$.