The breadth of an element $x$ in a finite $p$-group G is defined to be that integer $b = br(x)$ such that $p ^b = |G : C_ G (x)|$, while the breadth $br(G)$ of $G$ is the supremum of $\{br_ G (x) | x \in G\}$.
In [ J.A. Gallian, On the breadth of a finite p -group, Math. Z. 126 (1972), 224–226.], it has been proved that if $c$ denotes the nilpotency class of $G$, then $ c < 1+(pb−1)/(p−1)$. Does the above result work for all values of $p$ an $b$? (Because it seems that it does not work for $D_8$ and $Q_8$ )