#6686115

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682658
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#3054	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1599 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)(6,11)(10,16)(12,14)(13,15)(17,18,24,29)(19,25,33,50)(20,34,36,54)(21,27,41,58)(22,42,23,28)(26,44,52,38)(30,51,59,72)(31,47,63,71)(32,49,43,48)(35,64,61,46)(37,65,67,76)(39,55,62,73)(40,57,45,56)(53,69,75,68)(60,74,77,80)(66,79,78,70);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19)(18,25)(20,35)(21,39)(22,32)(23,43)(24,33)(26,51)(27,55)(28,48)(29,50)(30,38)(31,40)(34,64)(36,61)(37,60)(41,62)(42,49)(44,59)(45,63)(46,54)(47,57)(52,72)(53,70)(56,71)(58,73)(65,74)(66,69)(67,77)(68,78)(75,79)(76,80);(17,20,23,44,24,36,22,38)(18,26,42,54,29,52,28,34)(19,30,43,35,33,59,32,61)(21,37,45,69,41,67,40,68)(25,46,49,72,50,64,48,51)(27,53,57,76,58,75,56,65)(31,60,39,66,63,77,62,78)(47,70,73,80,71,79,55,74);(2,7)(6,9)(8,15)(14,16)(20,36)(26,52)(30,59)(34,54)(35,61)(37,67)(38,44)(46,64)(51,72)(53,75)(60,77)(65,76)(66,78)(68,69)(70,79)(74,80);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,21)(18,27)(19,31)(20,37)(22,40)(23,45)(24,41)(25,47)(26,53)(28,56)(29,58)(30,60)(32,62)(33,63)(34,65)(35,66)(36,67)(38,68)(39,43)(42,57)(44,69)(46,70)(48,55)(49,73)(50,71)(51,74)(52,75)(54,76)(59,77)(61,78)(64,79)(72,80)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6593311);