#486776

order	512
nilpotency class	4
isoclinism	isoclinic class of #486320
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

647 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)(19,33,38,63)(20,39,43,60)(21,29,45,57)(22,28,44,58)(23,31,24,30)(26,49,53,36)(27,47,56,42)(34,64,69,79)(35,52,46,61)(37,51,48,59)(40,54,73,78)(41,71,72,55)(50,74,77,65)(62,68,80,67)(66,70,75,76);(1,2)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(17,19,25,38)(18,26,32,53)(20,34,43,69)(21,36,45,49)(22,46,44,35)(23,37,24,48)(27,50,56,77)(28,33,58,63)(29,59,57,51)(30,52,31,61)(39,70,60,76)(40,66,73,75)(41,74,72,65)(42,67,47,68)(54,62,78,80)(55,79,71,64);(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)(23,47)(24,42)(25,43)(26,50)(28,54)(29,55)(30,60)(31,39)(32,56)(33,62)(35,65)(36,66)(37,68)(38,69)(44,72)(45,73)(46,74)(48,67)(49,75)(51,64)(52,76)(53,77)(57,71)(58,78)(59,79)(61,70)(63,80);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(17,21,24,44,25,45,23,22)(18,28,31,57,32,58,30,29)(19,35,37,36,38,46,48,49)(20,40,42,72,43,73,47,41)(26,51,52,33,53,59,61,63)(27,54,39,71,56,78,60,55)(34,65,68,66,69,74,67,75)(50,64,76,62,77,79,70,80)

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=dgi,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

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393972);