Kolmogorov operator with the vector field in Nash class

We establish sharp two-sided heat kernel bounds, Harnack inequality and H\"{o}lder continuity of bounded solutions for divergence-form parabolic equation with measurable uniformly elliptic matrix and the first-order term in a large class of locally unbounded vector fields containing $L^p$, $p>d$ as well as some vector fields $\not\in L_{loc}^p$, $p>2$.