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Let  K  be any field and  G  be a finite group acting on the rational function field  K ( x   g   :  g  ∈  G ) by  h  ⋅  x   g   =  x   hg   for any  g ,  h  ∈  G . Define  K ( G ) =  K ( x   g   :  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  p  6  with  B  0 ( G ) ≠ 0, where  B  0 ( G ) is the unramified Brauer group of ℂ( G ), which is the subgroup of  H  2 ( 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  B  0 ( G ) ≠ 0; for groups  G  satisfying  B  0 ( G ) = 0, we will show that ℂ( G ) is rational except possibly for five cases.

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