#489598

order	512
nilpotency class	4
isoclinism	isoclinic class of #489385
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	#62344	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

679 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)(3,11)(4,10)(5,8)(6,9)(7,13)(12,15)(14,16)(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)(2,7)(4,14)(5,13)(6,12)(8,15)(9,16)(10,11)(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);(2,8)(4,5)(7,15)(9,10)(11,16)(13,14)(17,20)(18,27)(19,34)(21,40)(22,41)(23,48)(24,42)(25,43)(26,51)(28,55)(29,56)(30,60)(31,39)(32,57)(33,62)(35,64)(36,65)(37,67)(38,68)(44,72)(45,73)(46,71)(47,75)(49,66)(50,76)(52,77)(53,78)(54,79)(58,80)(59,74)(61,70)(63,69);(2,9)(3,12)(8,10)(13,14)(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,d^2=gh,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fhi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fhi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396794);