#7624720

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624150
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

1181 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)(22,40)(23,44)(24,29)(26,37)(28,42)(30,55)(31,47)(32,48)(33,61)(34,64)(36,67)(38,68)(39,54)(41,50)(43,66)(45,53)(46,58)(49,63)(51,70)(52,69)(56,77)(57,78)(59,74)(60,76)(62,71)(65,75)(72,79)(73,80);(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)(26,46)(27,47)(28,49)(29,48)(35,55)(36,56)(37,58)(38,57)(39,59)(40,61)(41,62)(42,63)(43,60)(44,64)(45,65)(50,71)(51,73)(52,72)(53,75)(54,74)(66,76)(67,77)(68,78)(69,79)(70,80);(17,20,24,38)(18,26,29,52)(19,30,32,57)(21,36,23,45)(22,41,43,70)(25,46,48,72)(27,50,44,51)(28,53,54,67)(31,56,34,65)(33,62,60,80)(35,66,68,40)(37,39,69,42)(47,71,64,73)(49,75,74,77)(55,76,78,61)(58,59,79,63);(3,10)(6,9)(11,16)(14,15)(17,21,24,23)(18,27,29,44)(19,31,32,34)(20,36,38,45)(22,39,43,42)(25,47,48,64)(26,50,52,51)(28,40,54,66)(30,56,57,65)(33,59,60,63)(35,67,68,53)(37,41,69,70)(46,71,72,73)(49,61,74,76)(55,77,78,75)(58,62,79,80);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(19,32)(25,48)(30,57)(31,34)(33,60)(46,72)(47,64)(49,74)(55,78)(56,65)(58,79)(59,63)(61,76)(62,80)(71,73)(75,77)

Polycyclic presentation:

2^9|a^2=f,c^2=i,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,g^c=gh,h^c=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531916);