In this talk I will present how to obtain the small-time heat kernel asymptotic in the setting of step-two groups, where an explicit formula of the heat kernel in the form of oscillatory integral is known. Consequently, a complete answer to the long-standing Gaveau--Brockett open problem is also provided by the Varadhan's formula. Furthermore, I will show the relation between the heat kernel with the normal geodesics. This talk is based on a joint work with Prof. Hong-Quan Li (Fudan University).