#6529913

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500489
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1576 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=h,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dh,g^c=gi,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, 6437109);