Isometry classes of simply connected nilmanifolds

We classify the isometry equivalence classes and determine the isometry groups of connected and simply connected Riemannian nilmanifolds on filiform Lie groups of arbitrary dimension and on five dimensional nilpotent Lie groups of nilpotency class >2. To achieve this classification we prove that up to one exceptional class the five dimensional non two-step nilmanifolds and the filiform nilmanifolds have isometry groups of the same (minimal) dimension as the nilmanifold. We give a detailed description of the exceptional case.