#499770

order	512
nilpotency class	4
isoclinism	isoclinic class of #499769
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1249 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:

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

Polycyclic presentation:

2^9|d^2=i,e^2=hi,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,g^c=gi,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, 406966);