  The Cartan group is the free nilpotent Lie group of rank 2 and step 3. We
consider the left-invariant sub-Riemannian problem on the Cartan group defined
by an inner product in the first layer of its Lie algebra. This problem gives a
nilpotent approximation of an arbitrary sub-Riemannian problem with the growth
vector $(2,3,5)$. In previous works we described a group of symmetries of the
sub-Riemannian problem on the Cartan group, and the corresponding Maxwell time
-- the first time when symmetric geodesics intersect one another. It is known
that geodesics are not globally optimal after the Maxwell time. In this work we
study local optimality of geodesics on the Cartan group. We prove that the
first conjugate time along a geodesic is not less than the Maxwell time
corresponding to the group of symmetries. We characterize geodesics for which
the first conjugate time is equal to the first Maxwell time. Moreover, we
describe continuity of the first conjugate time near infinite values.
