#501271

order	512
nilpotency class	4
isoclinism	isoclinic class of #501270
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24424
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#61106	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2955 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1030 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:

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

Polycyclic presentation:

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

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, 408467);