  For a real, non-singular, 2-step nilpotent Lie algebra $\mathfrak{n}$, the
group \Aut(\mathfrak{n})/\Aut_0(\mathfrak{n})$, where $\Aut_0(\mathfrak{n})$ is
the group of automorphisms which act trivially on the center, is the direct
product of a compact group with the 1-dimensional group of dilations.
Maximality of some automorphisms groups of $\mathfrak{n}$ follows and is
related to how close is $\mathfrak{n}$ to being of Heisenberg type. For
example, at least when the dimension of the center is two, $\dim
\Aut(\mathfrak{n})$ is maximal if and only if $\mathfrak{n}$ is type $H$. The
connection with fat distributions is discussed.
