#486684

order	512
nilpotency class	4
isoclinism	isoclinic class of #486327
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	#62279	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

684 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=g,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,d^c=di,e^c=eh,f^c=fi

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, 393880);