Modular Heisenberg group ($H_p=U{T}_3({\mathbb{F}}_p)$)

The modular Heisenberg group consists of upper-unitriangular $3\times 3$ matrices over the finite field ${\mathbb{F}}_p$. In coordinates $M(x,y,z)$, multiplication is $M(x,y,z)M(x\prime ,y\prime ,z\prime )=M(x+x\prime ,y+y\prime +xz\prime ,z+z\prime )$, and its order is $p^3$. Its center of a group is $\{M(0,y,0)\}\cong C_p$; the quotient group by the centre is $C_p\times C_p$, represented by the $x,z$ coordinates. The elementary generators $a=M(1,0,0)$, $b=M(0,0,1)$, $c=M(0,1,0)$ satisfy $a^p=b^p=c^p=e$, $ab{a}^{-1}{b}^{-1}=c$, and $c$ central. For $p=2$, $(ab{)}^2=c$, so the whole group does not have exponent two.