The extended genus of a nilpotent groupNis the set of isomorphism classes of nilpotent groupsM,not necessarily finitely generated, such that thep-localizationsMp,Npare isomorphic for all primesp. In this article, for any torsion-free finitely generated nilpotent groupNof nilpotency class 2, the extended genus ofNis analyzed by assigning to each of its members a sequence of triads of matrices with rational entries, generalizing the sequential representation which has been exploited elsewhere in the case whenNis abelian. This approach allows, among other things, to obtain examples of groups in the ordinary (Mislin) genus ofN