  51 Eri is well known for hosting a directly-imaged giant planet and for its
membership to the $\beta$ Pictoris moving group. Using two-minute cadence
photometry from the Transiting Exoplanet Survey Satellite (TESS), we detect
multi-periodic variability in 51 Eri that is consistent with pulsations of
Gamma Doradus ($\gamma$ Dor) stars. We identify the most significant pulsation
modes (with frequencies between $\sim$0.5-3.9 cycles/day and amplitudes ranging
between $\sim$1-2 mmag) as dipole and quadrupole gravity-modes, as well as
Rossby modes, as previously observed in Kepler $\gamma$ Dor stars. Our results
demonstrate that previously reported variability attributed to stellar rotation
is instead likely due to $\gamma$ Dor pulsations. Using the mean frequency of
the $\ell =1$ gravity-modes, together with empirical trends of the Kepler
$\gamma$ Dor population, we estimate a plausible stellar core rotation period
of 0.9$^{+0.3}_{-0.1}$ days for 51 Eri. We find no significant evidence for
transiting companions around 51 Eri in the residual light curve. The detection
of $\gamma$ Dor pulsations presented here, together with follow-up observations
and modeling, may enable the determination of an asteroseismic age for this
benchmark system. Future TESS observations would allow a constraint on the
stellar core rotation rate, which in turn traces the surface rotation rate, and
thus would help clarify whether or not the stellar equatorial plane and orbit
of 51 Eri b are coplanar.
