  Classifying perturbative fixed points near upper critical dimensions plays an
important role in understanding the space of conformal field theories and
critical phases of matter. In this work, we consider perturbative fixed points
of $N=5$ scalar bosons coupled with quartic interactions preserving an
arbitrary subgroup $G\subset {\rm O}(5)$. We perform an exhaustive algorithmic
search over the symmetry groups $G$ which are irreducible and satisfy the
Landau condition, so that the fixed point can be reached by fine-tuning a
single mass term and there is no need to tune the cubic couplings. We also
impose stability of the RG flow in the space of quartic couplings, and reality.
We thus prove that there exist no new stable fixed points in $d=4-\epsilon$
dimensions beyond the two known ones: namely the ${\rm O}(5)$ invariant fixed
point and the Cubic(5) fixed point. This work is a continuation of the
classification of such fixed points with $N=4$ scalars by Toledano, Michel,
Toledano, and Br\'ezin in 1985.
