#485036

order	512
nilpotency class	4
isoclinism	isoclinic class of #484217
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	C₂×#41	C₂×D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2922 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

679 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:

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

Polycyclic presentation:

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

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, 392232);