Structure of the Macdonald groups in one parameter

Consider the Macdonald groups $G(\alpha )=⟨A,B\mathrm{\mid }{A}^{[A,B]}=A^{\alpha },B^{[B,A]}=B^{\alpha }⟩$, $\alpha \in \mathbf{Z}$. We fill a gap in Macdonald’s proof that $G(\alpha )$ is always nilpotent, and proceed to determine the order, upper and lower central series, nilpotency class, and exponent of $G(\alpha )$.