#7625048

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624824
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 ◁ #529 ◁ #2966 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1147 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 80 points:

(1,2,5,8)(3,9,10,6)(4,12,13,7)(11,14,16,15)(17,18)(19,25)(20,35)(21,27)(22,42)(23,44)(24,29)(26,38)(28,36)(30,52)(31,46)(32,47)(33,60)(34,61)(37,62)(39,68)(40,64)(41,51)(43,69)(45,56)(48,53)(49,66)(50,67)(54,73)(55,74)(57,79)(58,72)(59,80)(63,65)(70,77)(71,78)(75,76);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,19)(18,25)(20,30)(21,31)(22,33)(23,34)(24,32)(26,45)(27,46)(28,48)(29,47)(35,52)(36,53)(37,54)(38,56)(39,57)(40,55)(41,58)(42,60)(43,59)(44,61)(49,70)(50,71)(51,72)(62,73)(63,75)(64,74)(65,76)(66,77)(67,78)(68,79)(69,80);(17,20,36,65,23,39,43,67,24,40,41,66,21,37,22,38)(18,26,42,62,27,49,51,64,29,50,69,68,44,63,28,35)(19,30,53,76,34,57,59,78,32,55,58,77,31,54,33,56)(25,45,60,73,46,70,72,74,47,71,80,79,61,75,48,52);(3,10)(6,9)(11,16)(14,15)(17,21,24,23)(18,27,29,44)(19,31,32,34)(20,37,40,39)(22,41,43,36)(25,46,47,61)(26,49,50,63)(28,42,51,69)(30,54,55,57)(33,58,59,53)(35,62,64,68)(38,66,67,65)(45,70,71,75)(48,60,72,80)(52,73,74,79)(56,77,78,76);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(19,32)(25,47)(30,55)(31,34)(33,59)(45,71)(46,61)(48,72)(52,74)(53,58)(54,57)(56,78)(60,80)(70,75)(73,79)(76,77)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532244);