We show that strictly abnormal geodesics arise in graded nilpotent Lie groups. We construct such a group, with a left invariant bracket-generating distribution, for which some Carnot geodesics are strictly abnormal and, in fact, not normal in any subgroup. In the 2-step case we also prove that these geodesics are always smooth. Our main technique is based on the equations for the normal and abnormal curves, that we derive (for any Lie group) explicitly in terms of the structure constants.