#7625077

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

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,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, 7532273);