#499741

order	512
nilpotency class	4
isoclinism	isoclinic class of #499736
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1263 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)(7,21)(8,26)(9,31)(10,33)(11,19)(12,18)(13,20)(14,23)(15,29)(16,32)(17,36)(22,50)(24,55)(25,45)(27,49)(28,59)(30,48)(34,44)(35,47)(37,46)(38,43)(39,58)(40,54)(41,66)(42,62)(51,56)(52,67)(53,63)(57,61)(60,65)(64,68);(1,2)(3,4)(5,7,13,25)(6,14,20,40)(8,22,30,53)(9,11,24,12)(10,34,37,32)(15,39,43,65)(16,19,44,18)(17,31,47,55)(21,46,45,33)(23,35,54,36)(26,56,48,41)(27,38,52,29)(28,60,62,58)(42,63,59,50)(49,68,67,57)(51,64,66,61);(2,4)(5,8,13,30)(6,15,20,43)(7,22,25,53)(9,27,24,52)(10,28,37,62)(11,38,12,29)(14,39,40,65)(16,41,44,56)(17,42,47,59)(18,48,19,26)(21,49,45,67)(23,51,54,66)(31,63,55,50)(32,58,34,60)(33,57,46,68)(35,64,36,61);(5,9)(6,16)(7,11)(8,27)(10,23)(12,25)(13,24)(14,19)(15,41)(17,45)(18,40)(20,44)(21,47)(22,38)(26,39)(28,51)(29,53)(30,52)(31,33)(32,36)(34,35)(37,54)(42,67)(43,56)(46,55)(48,65)(49,59)(50,68)(57,63)(58,61)(60,64)(62,66)

Polycyclic presentation:

2^9|b^2=i,c^2=i,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,h^a=hi,c^b=cg,e^b=eh,h^b=hi,f^c=fi,e^d=eh,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, 406937);