Let k be any field,G be a finite group acting on the rational function field k (xg: g ∈G) by h·xg = x hg for any h,g G.Define k (G) = k (xg: g∈ 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 associated to G,which is a subgroup of H2 (G,Q/Z).Bogomolov proves that for any prime number p,there is a k -group G of order k 6 such that B0 (G) is non-trivial and therefore C(G) is not rational over C.He also shows that,if G is a k -group of order k5,then B0(G) = 0.The latter result was disproved by Moravec for k = 3,5,7 by the computer computing.The case for groups of order 32 and 64 was solved by Chu,Hu,Kang,Kunyvskii and Prokhorov.