#7625049

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

Polycyclic presentation:

2^9|a^2=f,c^2=ghi,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, 7532245);