#7624776

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
prime radical	#61477	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #529 ◁ #2966 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1179 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)(17,18,22,28)(19,30,33,25)(20,36,38,71)(21,27,24,45)(23,42,41,72)(26,50,52,39)(29,40,54,44)(31,59,61,69)(32,57,35,48)(34,65,64,74)(37,70,46,55)(43,51,73,53)(47,62,56,78)(49,67,58,63)(60,77,68,79)(66,76,80,75);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,19,22,33)(18,25,28,30)(20,31,38,61)(21,32,24,35)(23,34,41,64)(26,47,52,56)(27,48,45,57)(29,49,54,58)(36,69,71,59)(37,60,46,68)(39,62,50,78)(40,63,44,67)(42,74,72,65)(43,66,73,80)(51,75,53,76)(55,77,70,79);(17,20)(18,26)(19,31)(21,37)(22,38)(23,43)(24,46)(25,47)(27,51)(28,52)(29,55)(30,56)(32,60)(33,61)(34,66)(35,68)(36,42)(39,44)(40,50)(41,73)(45,53)(48,75)(49,77)(54,70)(57,76)(58,79)(59,65)(62,67)(63,78)(64,80)(69,74)(71,72);(3,10)(6,9)(11,16)(14,15)(17,21,22,24)(18,27,28,45)(19,32,33,35)(20,37,38,46)(23,40,41,44)(25,48,30,57)(26,51,52,53)(29,42,54,72)(31,60,61,68)(34,63,64,67)(36,70,71,55)(39,43,50,73)(47,75,56,76)(49,74,58,65)(59,77,69,79)(62,66,78,80);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(18,28)(19,33)(26,52)(27,45)(29,54)(31,61)(32,35)(34,64)(36,71)(42,72)(51,53)(55,70)(60,68)(62,78)(63,67)(66,80)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=i,e^2=fi,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, 7531972);