On Odd Order Nilpotent Groups With Class 2

Let $G$ be an odd order nilpotent group with class 2 and $e$ denotes the exponent of its commutator subgroup. Let $e=p_1^{r_1}{p}_2^{r_2}\mathrm{.}\mathrm{.}\mathrm{.}{p}_s^{r_s}$, where $p_i$’s are odd primes and $r_i$’s are non-negative integers. Then there are at least $r_1+r_2+\mathrm{.}\mathrm{.}\mathrm{.}+r_s$ non-isomorphic nilpotent groups with class two and the order of each of the group is equal to the order of $G$.