  Quadratic conjecture is a strengthening of oliver's $p$-group conjecture. Let
$G$ be a $p$-group of maximal class of order $p^n$. We prove that if $n\le 8$
or $n\ge \max\{2p-6,p+2\}$ then $G$ satisfies Quadratic Conjecture. Hence
quadratic conjecture holds if $G$ is a $p$-group of maximal class where $p\le
7$.
