  We describe the centers of the universal enveloping algebras of nilpotent Lie
algebras of dimension at most six over fields of prime characteristic. If the
characteristic is not smaller than the nilpontency class, then the center is
the integral closure of the algebra generated over the $p$-center by the same
generators that also occur in characteristic zero. Except for three examples
(two of which are standard filiform), this algebra is already integrally closed
and hence it coincides with the center. In the case of these three exceptional
algebras, the center has further generators. Then we show that the center of
the universal enveloping algebra of the algebras investigated in this paper is
isomorphic to the Poisson center (the algebra of invariants under the adjoint
representation). This shows that Braun's conjecture is valid for this class of
Lie algebras.
