  We prove that the bimeromorphic class of a hyperk\"ahler manifold deformation
equivalent to O'Grady's six dimensional one is determined by the Hodge
structure of its Beauville-Bogomolov lattice by showing that the monodromy
group is maximal. As applications, we give the structure for the K\"ahler and
the birational K\"ahler cones in this deformation class and we prove that the
existence of a square zero divisor implies the existence a rational lagrangian
fibration with fixed fibre types.
