#488714

order	512
nilpotency class	4
isoclinism	isoclinic class of #488707
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	#62274	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

715 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)(3,6)(4,7)(5,8)(9,15)(10,13)(11,14)(12,16)(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);(1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,16)(14,15)(17,19)(18,26)(20,34)(21,36)(22,47)(23,37)(24,49)(25,38)(27,51)(28,58)(29,52)(30,53)(31,61)(32,54)(33,46)(35,44)(39,70)(40,65)(41,75)(42,66)(43,68)(45,50)(48,67)(55,80)(56,77)(57,79)(59,69)(60,78)(62,71)(63,74)(64,72)(73,76);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,16)(13,15)(17,20,25,43)(18,27,32,57)(19,34,38,68)(21,40,45,73)(22,41,44,72)(23,48,24,42)(26,51,54,79)(28,55,46,71)(29,56,59,74)(30,60,31,39)(33,62,58,80)(35,64,47,75)(36,65,50,76)(37,67,49,66)(52,77,69,63)(53,78,61,70);(2,8)(6,13)(7,14)(12,15)(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,c^2=i,d^2=g,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dgh,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

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 395910);