Bogomolov multipliers of some groups of order $p^{6}$

Abstract

Let $G$ be a finite group, $V$ a faithful finite-dimensional representation of $G$ over the complex field $\mathbb{C}$ and $\mathbb{C}(V)^{G}$ be the corresponding invariant field. The Bogomolov multiplier $B_{0}(G)$ of $G$ is canonically isomorphic to the unramified cohomological group $H_{\textrm{nr}}^{2}(\mathbb{C}(V)^{G},\mathbb{Q}/\mathbb{Z})$, which has been used by Saltman (1984) and Bogomolov (1988) to provide counter-examples to the rationality problem of $\mathbb{C}(V)^{P}$ for finite $p$-groups $P$ over $\mathbb{C}$. In this paper, we investigate the vanishing property of $B_{0}(P)$, where $P$ denotes a $p$-group of order $p^{6}$ for $p\geqslant 3$.