A study of the Wamsley group and its Sylow subgroups

We study the Wamsley group $⟨X,Y,Z\mathrm{\mid }{X}^Z=X^{\alpha },{}^{Z}Y=Y^{\beta },Z^{\gamma }=[X,Y]⟩$ and its Sylow subgroups, where ${\alpha }^{\gamma }\mathrm{\ne }1\mathrm{\ne }{\beta }^{\gamma }$ and $\gamma >0$, obtaining the sharpest results when $\alpha =\beta$.