#6867940

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667676
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1906 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,d^2=hi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,g^c=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72305295, #72305296, #72305299
maximal normal subgroup of	#72305295, #72305296, #72305299
maximal quotient of	#72305527, #72305550, #72316029

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6775136);