Sylow subgroups of the Macdonald group on 2 parameters

Consider the Macdonald group $G(\alpha ,\beta )=⟨A,B\mathrm{\mid }{A}^{[A,B]}=A^{\alpha },B^{[B,A]}=B^{\beta }⟩$, where $\alpha$ and $\beta$ are integers different from one. We fill a gap in Macdonald’s original proof that $G(\alpha ,\beta )$ is nilpotent, and find the order and nilpotency class of each Sylow subgroup of $G(\alpha ,\beta )$.