Finding a finite $p$-group of nilpotency class $n$ for each $n>1$

Question

There's a problem in Rotman's group theory book that goes

For $n\ge1$, let $G_n$ be a finite $p$-group of class $n$. Define $H$ to be the group of sequences $(g_1,g_2,\dots)$ for $g_n\in G_n$ and $g_n=1$ for large $n$, $\dots$

I don't have an issue answering the question, so I'm not even putting the rest of it. But my question is, how do we know there exists a finite $p$-group of class $n$ for each $n$?

Answer 1

Since all finite $p$-groups are nilpotent, we can look for some series of nilpotent $p$-groups. The dihedral groups $D_{2^k}$ are $2$-groups of nilpotency class $k$ for all $k\ge 2$. We can also consider certain matrix groups over the field $\mathbb{F}_p$, namely the groups $UT(n,p)$ of unitriangular matrices of size $n$. These are the $p$-Sylow subgroups of $GL(n,p)$. The groups $UT(n,p)$ have nilpotency class $n-1$.