  The recently realized Bose-Einstein condensate of Dysprosium will become a
spin-8 spinor condensate at ultralow magnetic fields. In such a high spin
condensate, many phases with different symmetries can exist. Among them the
most interesting ones are those with non-abelian point group symmetry. In this
letter we discuss the variety of symmetry phases in a spin-8 condensate
resulting from numerical solutions of the Hamiltonian. We show that these
symmetries can be determined uniquely from the measurements of density
population on each spin component in an ultralow magnetic field, together with
the measurements of the collective modes in the zero-field limit. This method
can also be applied to Bose-Einstein condensate of other magnetic atoms, such
as Cr and Er.
