Birational classification of fields of invariants for groups of order 128

Let G   be a finite group acting on the rational function field C(xg:g∈G)C(xg:g∈G) by CC-automorphisms h(xg)=xhgh(xg)=xhg for any g,h∈Gg,h∈G. Noether's problem asks whether the invariant field C(G)=k(xg:g∈G)GC(G)=k(xg:g∈G)G is rational (i.e. purely transcendental) over CC. By Fischer's theorem, C(G)C(G) is rational over CC when G is a finite abelian group. Saltman and Bogomolov, respectively, showed that for any prime p there exist groups G   of order p9p9 and of order p6p6 such that C(G)C(G) is not rational over CC by showing the non-vanishing of the unramified Brauer group: Brnr(C(G))≠0Brnr(C(G))≠0, which is an avatar of the birational invariant H3(X,Z)torsH3(X,Z)tors given by Artin and Mumford where X   is a smooth projective complex variety whose function field is C(G)C(G). For p=2p=2, Chu, Hu, Kang and Prokhorov proved that if G   is a 2-group of order ≤32, then C(G)C(G) is rational over CC. Chu, Hu, Kang and Kunyavskii showed that if G   is of order 64, then C(G)C(G) is rational over CC except for the groups G   belonging to the two isoclinism families Φ13Φ13 with Brnr(C(G))=0Brnr(C(G))=0 and Φ16Φ16 with Brnr(C(G))≃C2Brnr(C(G))≃C2. Bogomolov and Böhning's theorem claims that if G1G1 and G2G2 belong to the same isoclinism family, then C(G1)C(G1) and C(G2)C(G2) are stably CC-isomorphic. We investigate the birational classification of C(G)C(G) for groups G   of order 128 with Brnr(C(G))≠0Brnr(C(G))≠0. Moravec showed that there exist exactly 220 groups G   of order 128 with Brnr(C(G))≠0Brnr(C(G))≠0 forming 11 isoclinism families ΦjΦj. We show that if G1G1 and G2G2 belong to Φ16Φ16, Φ31Φ31, Φ37Φ37, Φ39Φ39, Φ43Φ43, Φ58Φ58, Φ60Φ60 or Φ80Φ80 (resp. Φ106Φ106 or Φ114Φ114), then C(G1)C(G1) and C(G2)C(G2) are stably CC-isomorphic with Brnr(C(Gi))≃C2Brnr(C(Gi))≃C2. Explicit structures of non-rational fields C(G)C(G) are given for each cases including also the case Φ30Φ30 with Brnr(C(G))≃C2×C2Brnr(C(G))≃C2×C2.