The exponents of $p$-groups of maximal class and their Schur multipliers

Abstract

Let $P$ be a $p$-group of maximal class of order $p^n$ where $p,n>2$. If $n>p+1$ then the exponent of $P$ is $p^k$, where $k$ is the least integer greater or equal to $(n-1)/(p-1)$. If $n=p+1$ then $P$ has exponent $p^2$. And if $n\le p$ then $P$ has exponent $p$ or $p^2$, with both possibilities arising. The Schur multiplier of $P$ has exponent at most $p^k$, where $k$ is the least integer greater than or equal to $(n-2)/(p-1)$.