  In this work large families of naturally graded nilpotent Lie algebras in
arbitrary dimension and characteristic sequence (n,q,1), with n odd, satisfying
the centralizer property, are given. This condtion constitutes a
generalization, for a nilpotent Lie agebra, of the structural properties
charactrizing the Lie algebra $Q_{n}$. By considering certain cohomological
classes of the space $H^{2}(\frak{g},\mathbb{C})$, it is shown that, with few
exceptions, the isomorphism classses of these algebras are given by central
extensions of $Q_{n}$ by $\mathbb{C}^{p}$ which preserve the nilindex and the
natural graduation.
