Abstract The role of torsion in the local and global description of supersymmetric M2‐branes with fluxes and parabolic monodromies is determined. The monodromy corresponds to a representation of the fundamental group of the base manifold into the parabolic subgroup of , the group of isotopy classes of area‐preserving diffeomorphisms. These are M2‐branes with a quantum discrete spectrum with finite multiplicity. The global description of these M2‐branes is given by twisted torus bundles with monodromy. Given a representation, they are classified by , or equivalently, by the coinvariants associated with the parabolic monodromy subgroup. Previous constructions are generalized in two different ways. The first one considers parabolic monodromies with . This makes it possible to identify torsion cycles of order greater than one. It is shown that there are well‐defined nilmanifolds in three, four, and five dimensions with nontrivial torsion cycles contained in the global description of these M2‐branes. The torsion is also manifest in the equivalence classes of M2‐brane bundles. The second one is the action of the torsion on the coinvariants of the base manifold, which is analyzed. Together with the flux condition, the torsion defines explicitly the number of coinvariants for a given monodromy.