$p$-group: the Sylow $p$-subgroups of $S_n$ or $GL_n(\mathbb{F}_p)$.

The former can be described as follows: split $n$ up into $\lfloor \frac{n}{p} \rfloor$ blocks of size $p$ and a remainder, and allow the permutations that only permute individual blocks. Then split the blocks themselves into "2-blocks" of size $p$ and allow permutations that permute the 2-blocks etc. The result is an iterated wreath product.

The latter can be described as the subgroup of upper-triangular matrices with all ones on the diagonal, e.g. a Heisenberg group.