#356487

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂²×C₄⋊C₄
prime radical	C₂³×D₄	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2018 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=g,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,c^b=ci,d^b=de,e^d=ei,h^d=hi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263683);