  In this Letter we study discrete symmetries of mean field manifolds of
condensates of F=2 cold atoms, and various unconventional quantum vortices.
Discrete quaternion symmetries result in two species of spin defects that can
only appear in integer vortices while {\em cyclic} symmetries are found to
result in a phase shift of $2\pi/3$ (or $4\pi/3$) and therefore 1/3- (or 2/3-)
quantum vortices in condensates. We also briefly discuss 1/3-quantum vortices
in condensates of trimers.
