C₃²×#550 #10596351

order	576
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃² · Schur multiplier (H₂) = C₂⁵×C₃

Aliases

	subgroup	G/H
center, socle	C₂²×C₃²	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ G

936 conjugacy classes of subgroups; only the 498 normal subgroups are shown.

normal subgroups only

This group has 936 conjugacy classes of subgroups — too many to draw. Only its 498 normal subgroups are shown.

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

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 4 generators on 18 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(576, 8233);