#485034

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	#61009	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2922 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

689 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)(19,31,35,59)(20,36,37,56)(21,40,44,54)(22,29,23,28)(25,47,50,33)(26,45,51,39)(27,42,55,43)(32,60,61,77)(34,65,46,53)(38,69,68,52)(41,57,72,49)(48,70,75,80)(58,64,79,63)(62,66,74,76)(67,73,78,71);(1,2)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(17,19)(18,25)(20,32)(21,41)(22,34)(23,46)(24,35)(26,48)(27,53)(28,49)(29,57)(30,50)(31,40)(33,43)(36,66)(37,61)(38,70)(39,63)(42,47)(44,72)(45,64)(51,75)(52,77)(54,59)(55,65)(56,76)(58,67)(60,69)(62,71)(68,80)(73,74)(78,79);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,15)(14,16)(17,20,24,37)(18,26,30,51)(19,32,35,61)(21,38,44,68)(22,45,23,39)(25,48,50,75)(27,52,55,69)(28,56,29,36)(31,58,59,79)(33,62,47,74)(34,64,46,63)(40,67,54,78)(41,70,72,80)(42,73,43,71)(49,76,57,66)(53,77,65,60);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(20,37)(26,51)(32,61)(36,56)(38,68)(39,45)(48,75)(52,69)(58,79)(60,77)(62,74)(63,64)(66,76)(67,78)(70,80)(71,73)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=h,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, 392230);