  We show that $n$ thermal fermionic alkaline-earth atoms in a flat-bottom trap
allow one to robustly implement a spin model displaying two symmetries: the
$S_n$ symmetry that permutes atoms occupying different vibrational levels of
the trap and the SU($N$) symmetry associated with $N$ nuclear spin states. The
high symmetry makes the model exactly solvable, which, in turn, enables the
analytic study of dynamical processes such as spin diffusion in this SU($N$)
system. We also show how to use this system to generate entangled states that
allow for Heisenberg-limited metrology. This highly symmetric spin model should
be experimentally realizable even when the vibrational levels are occupied
according to a high-temperature thermal or an arbitrary non-thermal
distribution.
