#486772

order	512
nilpotency class	4
isoclinism	isoclinic class of #486312
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#466	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

649 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 80 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=gh,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393968);