  Let $\Gamma=(V,E)$ be a graph. The square graph $\Gamma^2$ of the graph
$\Gamma$ is the graph with the vertex set $V(\Gamma^2)=V$ in which two vertices
are adjacent if and only if their distance in $\Gamma$ is at most two. The
square graph of the hypercube $Q_n$ has some interesting properties. For
instance, it is highly symmetric and panconnected.
  In this paper, we investigate some algebraic properties of the graph
${Q^2_n}$. In particular, we show that the graph ${Q^2_n}$ is
distance-transitive. We show that the graph ${Q^2_n}$ is an imprimitive
distance-transitive graph if and only if $n$ is an odd integer. Also, we
determine the spectrum of the graph $Q_n^2$. Finally, we show that when $n >2$
is an even integer, then ${Q^2_n}$ is an automorphic graph, that is, $Q_n^2$ is
a distance-transitive primitive graph which is not a complete or a line graph.
