  We study critical points of $F(4)$ gauged supergravity in six dimensions
coupled to three vector multiplets. Scalar fields are described by
$\mathbb{R}^+\times \frac{SO(4,3)}{SO(4)\times SO(3)}$ coset space, and the
gauge group is given by $SO(3)_R\times SO(3)$ with $SO(3)_R$ being the
R-symmetry. The maximally supersymmetric critical point with all scalars
vanishing preserves the full $SO(3)_R\times SO(3)$ symmetry. This is dual to a
superconformal field theories (SCFT$_5$) arising from a near horizon geometry
of the D4-D8 brane system in type I$'$ theory with an enhanced global symmetry
$E_1\sim SU(2)$. Apart from this trivial critical point, we identify a new
supersymmetric critical point preserving the full supersymmetry with the
$SO(3)_R\times SO(3)$ symmetry broken to its diagonal subgroup. This critical
point should correspond to a new SCFT in five dimensions. We study an RG flow
solution interpolating between the SCFT with $E_1$ symmetry and the new
supersymmetric critical point. The flow describes a supersymmetric deformation
driven by a vacuum expectation value of relevant operators of dimension $3$. We
identify the dual operators with the mass terms for hypermultiplet scalars in
the dual field theory. The solution provides an example of analytic
supersymmetric RG flows in AdS$_6$/CFT$_5$ correspondence.
