#7623783

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623613
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #532 ◁ #2931 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1489 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 72 points:

(1,2,4,6)(3,7,8,5)(9,10,14,21)(11,19,26,17)(12,28,31,61)(13,23,16,20)(15,36,35,49)(18,43,46,32)(22,34,50,38)(24,53,29,62)(25,41,33,47)(27,59,58,67)(30,64,39,51)(37,52,66,45)(40,55,44,63)(42,57,48,65)(54,72,56,71)(60,70,68,69);(1,3)(2,5)(4,8)(6,7)(9,11,14,26)(10,17,21,19)(12,29,31,24)(13,33,16,25)(15,27,35,58)(18,44,46,40)(20,47,23,41)(22,42,50,48)(28,53,61,62)(30,54,39,56)(32,63,43,55)(34,65,38,57)(36,67,49,59)(37,68,66,60)(45,69,52,70)(51,72,64,71);(9,12,14,31)(10,18,21,46)(11,24,26,29)(13,30,16,39)(15,37,35,66)(17,40,19,44)(20,45,23,52)(22,51,50,64)(25,54,33,56)(27,60,58,68)(28,49,61,36)(32,34,43,38)(41,69,47,70)(42,71,48,72)(53,67,62,59)(55,65,63,57);(3,8)(5,7)(9,11)(10,19)(12,24)(13,33)(14,26)(15,27)(16,25)(17,21)(18,44)(20,41)(22,48)(23,47)(28,53)(29,31)(30,56)(32,55)(34,65)(35,58)(36,59)(37,60)(38,57)(39,54)(40,46)(42,50)(43,63)(45,69)(49,67)(51,72)(52,70)(61,62)(64,71)(66,68);(9,13,14,16)(10,20,21,23)(11,25,26,33)(12,30,31,39)(15,34,35,38)(17,41,19,47)(18,45,46,52)(22,49,50,36)(24,54,29,56)(27,57,58,65)(28,51,61,64)(32,37,43,66)(40,69,44,70)(42,67,48,59)(53,71,62,72)(55,68,63,60)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gh,h^a=hi,c^b=ci,d^b=df,e^b=ei,g^c=gh,h^c=hi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530979);