  A notable class of superconformal theories (SCFTs) in six dimensions is
parameterized by an integer $N$, an ADE group $G$, and two nilpotent elements
$\mu_\mathrm{L,R}$ in $G$. Nilpotent elements have a natural partial ordering,
which has been conjectured to coincide with the hierarchy of
renormalization-group flows among the SCFTs. In this paper we test this
conjecture for $G=\mathrm{SU}(k)$, where AdS$_7$ duals exist in IIA. We work
with a seven-dimensional gauged supergravity, consisting of the gravity
multiplet and two $\mathrm{SU}(k)$ non-Abelian vector multiplets. We show that
this theory has many supersymmetric AdS$_7$ vacua, determined by two nilpotent
elements, which are naturally interpreted as IIA AdS$_7$ solutions. The BPS
equations for domain walls connecting two such vacua can be solved
analytically, up to a Nahm equation with certain boundary conditions. The
latter admit a solution connecting two vacua if and only if the corresponding
nilpotent elements are related by the natural partial ordering, in agreement
with the field theory conjecture.
