C₂×C₃²×#509 #11770461

order	1152
nilpotency class	4
isoclinism	isoclinic class of #509
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃² · Schur multiplier (H₂) = C₂⁵×C₃

Aliases

C₂×C₃²×#509

	subgroup	G/H
center, socle	C₂²×C₃²	C₂×D₈
commutator, Frattini, cosocle	C₈	C₂⁴×C₃²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₂²×C₃²×C₄ ◁ C₂²×C₃²×C₈ ◁ G

1200 conjugacy classes of subgroups.

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 4 generators on 24 points:

(1,2)(19,20,21);(3,4)(5,8)(6,15)(7,9)(10,12)(11,13)(14,16)(17,18)(22,23,24);(4,8)(5,12)(6,13)(9,16)(10,11)(14,18)(15,17);(4,9)(8,16)(10,17)(11,15)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 4472);