#7623573

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623326
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	#3058	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1239 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)(3,9)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18)(19,25)(20,33)(21,27)(22,41)(23,45)(24,29)(26,36)(28,43)(30,56)(31,47)(32,55)(34,54)(35,61)(37,62)(38,51)(39,52)(40,69)(42,46)(44,60)(48,64)(49,66)(50,59)(53,67)(57,77)(58,75)(63,76)(65,79)(68,74)(70,71)(72,80)(73,78);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19,24,23)(18,25,29,45)(20,34,37,30)(21,38,39,31)(22,32,44,43)(26,48,50,46)(27,51,52,47)(28,41,55,60)(33,54,62,56)(35,57,65,63)(36,64,59,42)(40,67,68,58)(49,71,73,72)(53,74,75,69)(61,77,79,76)(66,70,78,80);(3,10)(6,9)(11,16)(13,15)(17,20,24,37)(18,26,29,50)(19,30,23,34)(21,35,39,65)(22,42,44,64)(25,46,45,48)(27,49,52,73)(28,54,55,56)(31,57,38,63)(32,59,43,36)(33,60,62,41)(40,70,68,80)(47,71,51,72)(53,76,75,77)(58,78,67,66)(61,74,79,69);(2,7)(6,9)(8,14)(13,15)(17,21)(18,27)(19,31)(20,35)(22,40)(23,38)(24,39)(25,47)(26,49)(28,53)(29,52)(30,57)(32,58)(33,61)(34,63)(36,66)(37,65)(41,69)(42,70)(43,67)(44,68)(45,51)(46,71)(48,72)(50,73)(54,76)(55,75)(56,77)(59,78)(60,74)(62,79)(64,80);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)(21,39)(27,52)(31,38)(35,65)(40,68)(47,51)(49,73)(53,75)(57,63)(58,67)(61,79)(66,78)(69,74)(70,80)(71,72)(76,77)

Polycyclic presentation:

2^9|b^2=i,c^2=i,e^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gh,h^a=hi,c^b=cfi,d^b=di,g^c=gh,h^c=hi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530769);