Let G be a finite group acting on the rational function field C(x(g) : g is an element of G) by C-automorphisms h(x(g)) = x(hg) for any g,h is an element of G. Noether's problem asks whether the invariant field C(G) = k(x(g) : g is an element of G)(G) is rational (i.e. purely transcendental) over C. By Fischer's theorem, C(G) is rational over C when G is a finite abelian group. Saltman and Bogomolov, respectively, showed that for any prime p there exist groups G of order p(9) and of order p(6) such that C(G) is not rational over C by showing the non-vanishing of the unraraified Brauer group: Bru(nr)(C(G)) not equal 0, which is an avatar of the birational invariant H-3 (X, Z)(tors) given by Artin and Mumford where X is a smooth projective complex variety whose function field is C(G). For p = 2, Chu, Hu, Kang and Prokhorov proved that if G is a 2-group of order <= 32, then C(G) is rational over C. Chu, Hu, Kang and Kunyavskii showed that if G is of order 64, then C(G) is rational over C except for the groups G belonging to the two isoclinism families Phi(13) with Br-nr(C(G)) = 0 and Phi(16) with Br-nr(C(G)) similar or equal to C-2. Bogomolov and Bohning's theorem claims that if G(1) and G(2) belong to the same isoclinism family, then C(G(1)) and C(G(2)) are stably C-isomorphic. We investigate the birational classification of C(G) for groups G of order 128 with Br-nr(C(G)) not equal 0. Moravec showed that there exist exactly 220 groups G of order 128 with Br-nr(C(G)) not equal 0 forming 11 isoclinism families Phi(j). We show that if G(1) and G(2) belong to Phi(16), Phi(31), Phi(37), Phi(39), Phi(43), Phi(58), Phi(60) or Phi(80) (resp. Phi(106) or Phi(114)), then C(G(1)) and C(G(2)) are stably C-isomorphic with Br-nr(C(G(i))) similar or equal to C-2. Explicit structures of non-rational fields C(G) are given for each cases including also the case Phi(30) with Br-nr(C(G)) similar or equal to C-2 X C-2. (C) 2015 Elsevier Inc. All rights reserved.