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Let   $G$   be a   $p$  -group of maximal class and order   $p^n$  . We determine whether or not the Bogomolov multiplier   ${\operatorname{B}}_0(G)$   is trivial in terms of the lower central series of   $G$   and   $P_1 = C_G(\gamma _2(G) / \gamma _4(G))$  . If in addition   $G$   has positive degree of commutativity and   $P_1$   is metabelian, we show how understanding   ${\operatorname{B}}_0(G)$   reduces to the simpler commutator structure of   $P_1$  . This result covers all   $p$  -groups of maximal class of large-enough order, and, furthermore, it allows us to give the first natural family of   $p$  -groups containing an abundance of groups with non-trivial Bogomolov multipliers. We also provide more general results on Bogomolov multipliers of   $p$  -groups of arbitrary coclass   $r$  .

BOGOMOLOV MULTIPLIERS OF P -GROUPS OF MAXIMAL CLASS