  We study the intrinsic shapes of a sample of over 400 quiescent galaxies in
the cores of the Virgo and Fornax clusters with luminosities $10^{6} \leq
L_{g}/L_{\odot} \leq 10^{8}$. Similar to satellites of the Local Group and
Centaurus A, these faint, low surface brightness cluster galaxies are best
described as a family of thick ($C/A > 0.5$), oblate-triaxial spheroids.
However, the large sample size allows us to show that the flattening of their
stellar distributions depends both on luminosity and on the presence of a
nuclear star cluster. Nucleated satellites are thicker at all luminosities
compared to their non-nucleated counterparts, and fainter galaxies are
systematically thicker as well, regardless of nucleation. Once nucleation is
accounted for, we find no evidence that the environment the satellites live in
plays a relevant role in setting their three-dimensional structure. We
interpret both the presence of stellar nuclei and the associated thicker shapes
as the result of preferential early and rapid formation, effectively making
these faint nucleated galaxies the first generation of cluster satellites.
