#7623136

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622825
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 ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1490 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)(3,6)(4,5)(7,8)(9,10)(11,17)(12,19)(13,20)(14,33)(15,37)(16,22)(18,28)(21,35)(23,39)(24,40)(25,56)(26,43)(27,47)(29,49)(30,42)(31,48)(32,50)(34,44)(36,62)(38,52)(41,55)(45,61)(46,59)(51,65)(53,68)(54,67)(57,66)(58,69)(60,70)(63,71)(64,72);(2,5)(6,8)(9,11,16,26)(10,17,22,43)(12,23,29,54)(13,30,15,24)(14,25,36,58)(18,38,46,64)(19,39,49,67)(20,42,37,40)(21,41,50,70)(27,53,31,51)(28,52,59,72)(32,60,35,55)(33,56,62,69)(34,57,61,71)(44,66,45,63)(47,68,48,65);(9,12)(10,18)(11,23)(13,27)(14,34)(15,31)(16,29)(17,38)(19,33)(20,44)(21,48)(22,46)(24,51)(25,57)(26,54)(28,35)(30,53)(32,59)(36,61)(37,45)(39,56)(40,63)(41,65)(42,66)(43,64)(47,50)(49,62)(52,55)(58,71)(60,72)(67,69)(68,70);(1,3,4,7)(2,6,5,8)(9,12)(10,19)(11,23)(13,31)(14,28)(15,27)(16,29)(17,39)(18,33)(20,48)(21,44)(22,49)(24,53)(25,52)(26,54)(30,51)(32,61)(34,35)(36,59)(37,47)(38,56)(40,68)(41,66)(42,65)(43,67)(45,50)(46,62)(55,57)(58,72)(60,71)(63,70)(64,69);(9,13,16,15)(10,20,22,37)(11,24,26,30)(12,27,29,31)(14,32,36,35)(17,40,43,42)(18,44,46,45)(19,47,49,48)(21,33,50,62)(23,51,54,53)(25,55,58,60)(28,34,59,61)(38,63,64,66)(39,65,67,68)(41,69,70,56)(52,71,72,57)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530332);