C₃²×#533 #10596334

order	576
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃² · Schur multiplier (H₂) = C₂⁴×C₃

Aliases

	subgroup	G/H
center	C₂×C₃²×C₄	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃²
socle	C₂²×C₃²	C₂⁴
prime radical	C₂³×C₃²	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃²×C₄ ◁ G

666 conjugacy classes of subgroups; only the 510 normal subgroups are shown.

normal subgroups only

This group has 666 conjugacy classes of subgroups — too many to draw. Only its 510 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 4 generators on 22 points:

(1,2,3)(7,8)(9,12)(10,11)(13,14)(15,16,18,20)(17,21,22,19);(4,5,6)(8,11)(12,14);(15,17,18,22)(16,19,20,21);(7,9,10,13)(8,12,11,14)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(576, 8216);