  We measure the full elastic tensors of Mn$_3$Ge and Mn$_3$Sn as a function of
temperature through their respective antiferromagnetic phase transitions. Large
discontinuities in the bulk moduli at the N\'eel transitions indicate strong
magnetoelastic coupling in both compounds. Strikingly, the discontinuities are
nearly a factor of 10 larger in Mn$_3$Ge than in Mn$_3$Sn. We use the
magnitudes of the discontinuities to calculate the pressure derivatives of the
N\'eel temperature, which are 39 K/GPa 14.3 K/GPa for Mn$_3$Ge and Mn$_3$Sn,
respectively. We measured in-plane shear modulus $c_{66}$, which couples
strongly to the magnetic order, in magnetic fields up to 18 T and found
quantitatively similar behavior in both compounds. Recent measurements have
demonstrated strong piezomagnetism in Mn$_3$Sn: our results suggest that
Mn$_3$Ge may be an even better candidate for this effect.
