#6686105

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682648
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	#3050	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)(19,25,31,47)(20,34,38,53)(21,27,41,58)(22,42,23,28)(26,44,54,37)(30,51,59,72)(32,48,62,71)(33,50,43,49)(35,64,61,46)(36,65,69,76)(39,55,63,73)(40,57,45,56)(52,68,75,67)(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)(22,33)(23,43)(24,31)(26,51)(27,55)(28,49)(29,47)(30,37)(32,40)(34,64)(36,60)(38,61)(41,63)(42,50)(44,59)(45,62)(46,53)(48,57)(52,70)(54,72)(56,71)(58,73)(65,74)(66,68)(67,78)(69,77)(75,79)(76,80);(17,20,23,44,24,38,22,37)(18,26,42,53,29,54,28,34)(19,30,43,35,31,59,33,61)(21,36,45,68,41,69,40,67)(25,46,50,72,47,64,49,51)(27,52,57,76,58,75,56,65)(32,60,39,66,62,77,63,78)(48,70,73,80,71,79,55,74);(2,7)(6,9)(8,15)(14,16)(19,31)(25,47)(30,59)(32,62)(33,43)(35,61)(39,63)(46,64)(48,71)(49,50)(51,72)(55,73)(60,77)(66,78)(70,79)(74,80);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,21)(18,27)(19,32)(20,36)(22,40)(23,45)(24,41)(25,48)(26,52)(28,56)(29,58)(30,60)(31,62)(33,63)(34,65)(35,66)(37,67)(38,69)(39,43)(42,57)(44,68)(46,70)(47,71)(49,55)(50,73)(51,74)(53,76)(54,75)(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,d^b=di,e^b=efh,h^b=hi,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, 6593301);