  In this work, we give a sufficient condition for a step-two Carnot group to
satisfy the quasi Bakry-\'Emery curvature condition. As an application, we
establish the gradient estimate for the heat semigroup on the free step-two
Carnot group with three generators $N_{3,2}$. Moreover, high order gradient
estimates and the Riemannian counterparts are also deduced under an extra
condition.
