#7624767

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

1191 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=fi,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, 7531963);