  Consider the Macdonald groups $G(\alpha)=\langle A,B\,|\,
A^{[A,B]}=A^\alpha,\, B^{[B,A]}=B^\alpha\rangle$, $\alpha\in{\mathbf Z}$. 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 $G(\alpha)$.
