Noether’s Problem and the Unramified Brauer Group for Groups of Order 64

Let K be any field and G be a finite group acting on the rational function field K(xg : g ∈ G) by h ⋅ xg = xhg for any g, h ∈ G. Define K(G) = K(xg : g ∈ G)^G. Noether’s problem asks whether K(G) is rational (purely transcendental) over K. For any prime number p, Bogomolov shows that there is some group G of order p⁶ with B0(G) ≠ 0, where B0(G) is the unramified Brauer group of ℂ(G), which is the subgroup of H²(G, ℚ/ℤ) consisting of cohomology classes whose restrictions to all bicyclic subgroups are zero. As a consequence, ℂ(G) is not rational over ℂ. In this paper, we will classify all the groups G of order 64 with B0(G) ≠ 0; for groups G satisfying B0(G) = 0, we will show that ℂ(G) is rational except possibly for five cases.