In this paper we study the sub-Riemannian geodesic problem determined by a left invariant distribution of dimension n in a nilpotent Lie group G of dimension n(n+1)/2. The problem is formulated as an left invariant optimal control sytem. We describe a group action that leaves invariant the system. We apply Pontryagin maximum to derive necessary conditions for the optimality of the geodesics. For n odd abnormal extremals do exists, but they turn out to be non-strictly abnormal. The adjoint equation for the normal extremals is explicitly integrated. At the end, we specialize our results in some of the known low dimensional cases, namely, the Heisenberg (2,3) case and the (3,6) case.