#501268

order	512
nilpotency class	4
isoclinism	isoclinic class of #501262
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#61051	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1055 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,e^2=h,f^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,h^b=hi,e^c=ei,f^c=fi,e^d=ehi,f^d=fi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408464);