#489311

order	512
nilpotency class	4
isoclinism	isoclinic class of #488901
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	C₄×C₈	C₂⁴
prime radical	#121	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

666 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:

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

Polycyclic presentation:

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

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, 396507);