Sylow subgroups of the Macdonald group on 2 parameters

Consider the Macdonald group G(α,β)= A,B\,|\, A[A,B]=A^α,\, B[B,A]=B^β, where α and β are integers different from one. We fill a gap in Macdonald's original proof that G(α,β) is nilpotent, and find the order and nilpotency class of each Sylow subgroup of G(α,β).