Balanced two-generator $p$-groups of deficiency zero and unbounded nilpotency class
For every prime $p>3$ we construct an explicit family of finite two-generator $p$-groups of deficiency zero whose nilpotency classes tend to infinity. The construction gives, for every $n\geq1$, a group of order $p^{2n+1}$ and class $n+1$. This gives an affirmative answer to Problem 17.113 of the Kourovka Notebook.