# Finite group generated by abelian normal subgroups may have arbitrarily large nilpotency class

It is possible to have a finite group (more specifically, a group of prime power order for any chosen prime number $p$) that has arbitrarily large nilpotency class but is a group generated by abelian normal subgroups.
The idea is to take the group $U(n,p)$ of upper triangular matrices with $1$s on the diagonal and entries in the prime field $\mathbb{F}_p$. This group has nilpotency class $n - 1$. Moreover, it is generated by abelian normal subgroups parameterized by top right rectangles: for every $k \times (n - k)$ top right rectangle, define the corresponding subgroup as the subgroup where the diagonal has 1s, entries in the rectangle are arbitrary, and all other entries are $0$. This subgroup is elementary abelian of order $p^{k(n - k)}$, and these subgroups together generate the whole group.