Noether's problem and unramified brauer groups

Let k be any field, G be a finite group acting on the rational function field k(x_g : g ε G) by h · x_g = x_hg for any h, g ε G. Define k(G) = k(x_g : g ε G)^G. Noether's problem asks whether k(G) is rational (= purely transcendental) over k. It is known that, if ℂ(G) is rational over ℂ, then B₀(G) = 0 where B₀(G) is the unramified Brauer group of ℂ(G) over ℂ. Bogomolov showed that, if G is a p-group of order p5, then B₀(G) = 0. This result was disproved by Moravec for p = 3, 5, 7 by computer calculations. We will prove the following theorem. Theorem. Let p be any odd prime number, G be a group of order p5. Then B₀(G) ≠ 0 if and only if G belongs to the isoclinism family π₁₀ in R. James's classification of groups of order p⁵.