#490220

order	512
nilpotency class	4
isoclinism	isoclinic class of #490099
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	#62325	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

693 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)(3,11)(4,9)(5,8)(6,10)(7,13)(12,15)(14,16)(17,18,25,32)(19,33,39,67)(20,40,44,65)(21,29,47,61)(22,28,46,62)(23,31,24,30)(26,51,56,36)(27,50,59,43)(34,68,72,80)(35,55,48,54)(37,63,38,53)(41,57,45,60)(42,64,49,58)(52,73,79,69)(66,75,77,71)(70,76,74,78);(1,3)(2,7)(4,14)(5,13)(6,12)(8,15)(9,11)(10,16)(17,19,25,39)(18,26,32,56)(20,34,44,72)(21,36,47,51)(22,48,46,35)(23,38,24,37)(27,52,59,79)(28,33,62,67)(29,63,61,53)(30,55,31,54)(40,76,65,78)(41,70,45,74)(42,73,49,69)(43,75,50,71)(57,77,60,66)(58,80,64,68);(2,8)(4,5)(7,15)(9,10)(11,16)(13,14)(17,20)(18,27)(19,34)(21,45)(22,49)(23,50)(24,43)(25,44)(26,52)(28,60)(29,64)(30,65)(31,40)(32,59)(33,66)(35,73)(36,74)(37,75)(38,71)(39,72)(41,47)(42,46)(48,69)(51,70)(53,80)(54,76)(55,78)(56,79)(57,62)(58,61)(63,68)(67,77);(3,12)(7,16)(11,15)(13,14)(17,21,24,46,25,47,23,22)(18,28,31,61,32,62,30,29)(19,35,38,36,39,48,37,51)(20,41,43,49,44,45,50,42)(26,53,55,33,56,63,54,67)(27,57,40,64,59,60,65,58)(34,69,71,70,72,73,75,74)(52,68,78,77,79,80,76,66)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397416);