On families of nilpotent subgroups and associated coset posets

We study some properties of the coset poset associated with the family of subgroups of class $\leq 2$ of a nilpotent group of class $\leq 3$. We prove that under certain assumptions on the group the coset poset is simply-connected if and only if the group is $2$-Engel, and $2$-connected if and only if the group is nilpotent of class $2$ or less. We determine the homotopy type of the coset poset for the group of $4\times 4$ upper unitriangular matrices over $\mathbb{F}_p$, and for the Burnside groups of exponent $3$.