#490106

order	512
nilpotency class	4
isoclinism	isoclinic class of #490099
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	#62432	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

687 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:

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

Polycyclic presentation:

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

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, 397302);