  We study the Rumin differentials of the 5-dimensional graded nilpotent Lie
group that appears as osculating group of generic rank two distributions in
dimension five. In irreducible unitary representations of this group, the Rumin
differentials provide intriguing generalizations of the classical quantum
harmonic oscillator. For the Schrodinger representations, we compute the
spectrum and the zeta regularized determinant of each Rumin differential. In
the generic representations, we evaluate their alternating product, i.e., the
analytic torsion of the Rumin complex.
