#509120

order	512
nilpotency class	4
isoclinism	isoclinic class of #509119
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#30400
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2955 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

644 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|b^2=h,d^2=i,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fh,h^a=hi,c^b=cg,d^b=de,g^b=gi,f^c=fh,h^c=hi,e^d=ei,f^d=fh,h^d=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416316);