  G\,29$-$38 (TIC~422526868) is one of the brightest ($V=13.1$) and closest ($d
= 17.51$\,pc) pulsating white dwarfs with a hydrogen-rich atmosphere (DAV/ZZ
Ceti class). It was observed by the {\sl TESS} spacecraft in sectors 42 and 56.
The atmosphere of G~29$-$38 is polluted by heavy elements that are expected to
sink out of visible layers on short timescales. The photometric {\sl TESS} data
set spans $\sim 51$ days in total, and from this, we identified 56 significant
pulsation frequencies, that include rotational frequency multiplets. In
addition, we identified 30 combination frequencies in each sector. The
oscillation frequencies that we found are associated with $g$-mode pulsations,
with periods spanning from $\sim$ 260 s to $\sim$ 1400 s. We identified %three
distinct rotational frequency triplets with a mean separation $\delta
\nu_{\ell=1}$ of 4.67 $\mu$Hz and a quintuplet with a mean separation $\delta
\nu_{\ell=2}$ of 6.67 $\mu$Hz, from which we estimated a rotation period of
about $1.35 \pm 0.1$ days. We determined a constant period spacing of 41.20~s
for $\ell= 1$ modes and 22.58\,s for $\ell= 2$ modes. We performed
period-to-period fit analyses and found an asteroseismological model with
$M_{\star}/M_{\odot}=0.632 \pm 0.03$, $T_{\rm eff}=11\, 635\pm 178$ K, and
$\log{g}=8.048\pm0.005$ (with a hydrogen envelope mass of $M_{\rm H}\sim
5.6\times 10^{-5}M_{\star}$), in good agreement with the values derived from
spectroscopy. We obtained an asteroseismic distance of 17.54 pc, which is in
excellent agreement with that provided by {\sl Gaia} (17.51 pc).
