SO(8)

	Group (mathematics)
Basic notions
	Subgroup; Normal subgroup; Quotient group; Group homomorphism; (semi-)direct product;
Discrete groups
	Classification of finite simple groups; Cyclic group Zn; Alternating group An; Sporadic groups; Mathieu group M11..12,M22..24; Conway group Co1..3; Janko group J1, 2, 3, 4; Fischer group F22..24; Baby Monster group B; Monster group M; Other finite groups; Symmetric group, Sn; Dihedral group, Dn; Infinite groups; The integers, Z; Modular groups, PSL(2,Z) and SL(2,Z);
Continuous groups
	Lie group; General linear group GL(n); Special linear group SL(n); Orthogonal group O(n); Special orthogonal group SO(n); Unitary group U(n); Special unitary group SU(n); Symplectic group Sp(n); G2 F4 E6 E7 E8; Lorentz group; Poincaré group;
Infinite dimensional groups
	conformal group; diffeomorphism group; Loop group; Quantum group; O(∞) SU(∞) Sp(∞);
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In mathematics, SO(8) is the special orthogonal group acting on eight-dimensional Euclidean space. It could be either a real or complex simple Lie group of rank 4 and dimension 28. Like all special orthogonal groups (except SO(2)), SO(8) is not simply connected, having a fundamental group isomorphic to Z₂. The universal cover of SO(8) is the spin group Spin(8). The center of Spin(8) is Z₂×Z₂ while the center of SO(8) is Z₂.

Dynkin diagram of SO(8), (D₄)

SO(8) is unique among the simple Lie groups in that its Dynkin diagram (shown right) (D₄ under the Dynkin classification) possesses a three-fold symmetry. This gives rise to peculiar feature of Spin(8) known as triality. Related to this is the fact that the two spinor representations, as well as the fundamental vector representation, of Spin(8) are all eight-dimensional (for all other spin groups the spinor representation is either smaller or larger than the vector representation). The triality automorphism of Spin(8) lives in the outer automorphism group of Spin(8) which is isomorphic to the symmetric group S₃ that permutes these three representations. The automorphism group acts on the center Z₂ x Z₂. When one quotients Spin(8) by one central Z₂, breaking this symmetry and obtaining SO(8), the remaining outer automorphism group is only Z₂. The triality symmetry acts again on the further quotient SO(8)/Z₂.