#7624719

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

1182 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)(36,66)(38,67)(39,54)(41,50)(43,69)(45,53)(46,58)(49,63)(51,70)(52,68)(56,76)(57,77)(59,74)(60,79)(62,71)(65,75)(72,78)(73,80);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,19,24,32)(18,25,29,48)(20,30,38,57)(21,31,23,34)(22,33,43,60)(26,46,52,72)(27,47,44,64)(28,49,54,74)(35,55,67,77)(36,56,45,65)(37,58,68,78)(39,59,42,63)(40,61,69,79)(41,62,70,80)(50,71,51,73)(53,75,66,76);(17,20)(18,26)(19,30)(21,36)(22,41)(23,45)(24,38)(25,46)(27,50)(28,53)(29,52)(31,56)(32,57)(33,62)(34,65)(35,40)(37,42)(39,68)(43,70)(44,51)(47,71)(48,72)(49,75)(54,66)(55,61)(58,63)(59,78)(60,80)(64,73)(67,69)(74,76)(77,79);(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)(28,40,54,69)(30,56,57,65)(33,59,60,63)(35,66,67,53)(37,41,68,70)(46,71,72,73)(49,61,74,79)(55,76,77,75)(58,62,78,80);(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)(55,77)(56,65)(58,78)(59,63)(61,79)(62,80)(71,73)(75,76)

Polycyclic presentation:

2^9|a^2=f,b^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, 7531915);