  Recently, extended gapless phases with emergent SU(2)$_1$ conformal
invariance occupying finite regions in the phase diagrams have been found in
one-dimensional spin-1/2 models with nonsymmorphic $O_h$ symmetry groups. In
this work, we investigate the question of whether the conditions for emergent
SU(2)$_1$ invariance can be loosened. We find that besides the nonsymmorphic
$O_h$ group, the other four smaller nonsymmorphic cubic groups including $O$,
$T_h$, $T_d$ and $T$ can also give rise to emergent SU(2)$_1$ invariance.
Minimal spin-1/2 models having these nonsymmorphic cubic groups as symmetry
groups are constructed, and numerical evidences for the emergent SU(2)$_1$
invariance are provided. Our work is useful for understanding gapless phases in
one-dimensional spin systems with nonsymmorphic symmetries.
