Calculating the genus of a direct product of certain nilpotent groups

The Mislin genus $\Cal G(N)$ of a finitely generated nilpotent group $N$ with finite commutator subgroup admits an abelian group structure. If $N$ satisfies some additional conditions ---we say that $N$ belongs to $\Cal N_1$--- we know exactly the structure of $\Cal G(N)$. Considering a direct product $ N_1 \times \cdots \times N_k$ of groups in $\Cal N_1$ takes us virtually always out of $\Cal N_1$. We here calculate the Mislin genus of such a direct product.