#501253

order	512
nilpotency class	4
isoclinism	isoclinic class of #501252
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)(3,15,19,41,21,46,20,48)(4,22,24,42,26,56,25,55)(5,27,30,11,32,43,31,44)(9,33,35,53,36,28,16,18)(10,37,38,57,40,29,39,23)(17,34,50,60,51,61,54,47)(45,52,59,58,63,64,62,49);(1,3)(2,9)(4,17)(5,28)(6,19)(7,20)(8,21)(10,34)(11,41)(12,35)(13,16)(14,36)(15,27)(18,30)(22,45)(23,52)(24,50)(25,54)(26,51)(29,49)(31,53)(32,33)(37,58)(38,60)(39,47)(40,61)(42,59)(43,46)(44,48)(55,62)(56,63)(57,64);(1,2)(3,16)(4,10)(5,11)(6,13)(7,12)(8,14)(9,20)(15,28)(17,47)(18,48)(19,36)(21,35)(22,23)(24,39)(25,38)(26,40)(27,30)(29,42)(31,43)(32,44)(33,46)(34,54)(37,55)(41,53)(45,49)(50,61)(51,60)(52,62)(56,57)(58,63)(59,64);(1,4,8,26)(2,10,14,40)(3,17,21,51)(5,29,32,37)(6,24,7,25)(9,34,36,61)(11,42,44,55)(12,38,13,39)(15,45,46,63)(16,47,35,60)(18,52,53,64)(19,50,20,54)(22,43,56,27)(23,31,57,30)(28,49,33,58)(41,59,48,62)

Polycyclic presentation:

2^9|a^2=f,d^2=i,e^2=hi,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=eh,g^b=gi,h^b=hi,f^c=fi,g^c=gi,e^d=eh,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, 408449);