#7625083

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624888
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	#62490	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #529 ◁ #2966 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1140 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)(17,18,24,29)(19,25,32,47)(20,35,38,65)(21,27,23,44)(22,42,43,69)(26,49,51,39)(28,41,53,36)(30,54,58,77)(31,46,34,63)(33,62,61,80)(37,64,40,66)(45,70,73,57)(48,60,74,55)(50,68,52,67)(56,75,59,76)(71,79,72,78);(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)(26,45)(27,46)(28,48)(29,47)(35,54)(36,55)(37,56)(38,58)(39,57)(40,59)(41,60)(42,62)(43,61)(44,63)(49,70)(50,71)(51,73)(52,72)(53,74)(64,75)(65,77)(66,76)(67,78)(68,79)(69,80);(17,20,36,67,23,37,43,49,24,38,41,68,21,40,22,39)(18,26,42,66,27,52,53,65,29,51,69,64,44,50,28,35)(19,30,55,78,34,56,61,70,32,58,60,79,31,59,33,57)(25,45,62,76,46,72,74,77,47,73,80,75,63,71,48,54);(3,10)(6,9)(11,16)(14,15)(17,21,24,23)(18,27,29,44)(19,31,32,34)(20,37,38,40)(22,41,43,36)(25,46,47,63)(26,50,51,52)(28,42,53,69)(30,56,58,59)(33,60,61,55)(35,64,65,66)(39,67,49,68)(45,71,73,72)(48,62,74,80)(54,75,77,76)(57,78,70,79);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(19,32)(20,38)(25,47)(26,51)(31,34)(33,61)(35,65)(37,40)(39,49)(46,63)(48,74)(50,52)(55,60)(62,80)(64,66)(67,68)

Polycyclic presentation:

2^9|a^2=fi,c^2=gh,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,d^c=di,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532279);