#7624768

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624170
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

1188 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)(20,34)(21,27)(22,40)(23,44)(24,29)(26,36)(28,42)(30,56)(31,47)(32,48)(33,63)(35,66)(37,67)(38,50)(39,55)(41,51)(43,65)(45,54)(46,59)(49,61)(52,70)(53,68)(57,75)(58,77)(60,80)(62,73)(64,76)(69,74)(71,78)(72,79);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,19)(18,25)(20,30)(21,38)(22,33)(23,31)(24,32)(26,46)(27,50)(28,49)(29,48)(34,56)(35,60)(36,59)(37,58)(39,69)(40,63)(41,64)(42,61)(43,62)(44,47)(45,57)(51,76)(52,71)(53,72)(54,75)(55,74)(65,73)(66,80)(67,77)(68,79)(70,78);(17,20,24,37)(18,26,29,53)(19,30,32,58)(21,35,23,45)(22,41,43,70)(25,46,48,72)(27,51,44,52)(28,54,55,66)(31,57,38,60)(33,64,62,78)(34,65,67,40)(36,39,68,42)(47,71,50,76)(49,75,74,80)(56,73,77,63)(59,69,79,61);(3,10)(6,9)(11,16)(14,15)(17,21,24,23)(18,27,29,44)(19,31,32,38)(20,35,37,45)(22,39,43,42)(25,47,48,50)(26,51,53,52)(28,40,55,65)(30,57,58,60)(33,61,62,69)(34,66,67,54)(36,41,68,70)(46,71,72,76)(49,73,74,63)(56,75,77,80)(59,78,79,64);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(19,32)(25,48)(30,58)(31,38)(33,62)(46,72)(47,50)(49,74)(56,77)(57,60)(59,79)(61,69)(63,73)(64,78)(71,76)(75,80)

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=dfi,e^b=ei,g^c=gh,h^c=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531964);