#501272

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
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2955 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1019 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:

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

Polycyclic presentation:

2^9|a^2=f,b^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, 408468);