  In this work, we establish the uniform heat kernel asymptotics as well as
sharp bounds for its derivatives on the free step-two Carnot group with $3$
generators. As a by-product, on this highly non-trivial toy model, we
completely solve the Gaveau-Brockett problem, in other words, we obtain the
expression of the squared Carnot-Carath\'eodory distance, as explicitly as one
can possibly hope for. Furthermore, the precise estimates of the heat kernel,
and its small-time asymptotic behaviors are deduced.
