#499775

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

1236 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)(7,20)(8,27)(9,33)(10,37)(11,18)(12,19)(13,43)(14,23)(15,31)(16,34)(17,41)(21,53)(22,46)(24,49)(25,56)(26,64)(28,55)(29,58)(30,66)(32,52)(35,65)(36,67)(38,54)(39,59)(40,57)(42,51)(44,45)(47,62)(48,63)(50,60)(61,68);(5,7,11,24)(6,14,18,47)(8,28,31,38)(9,22,35,60)(10,29,39,63)(12,26,13,25)(15,33,27,65)(16,46,53,50)(17,56,57,64)(19,58,43,48)(20,59,49,37)(21,42,34,32)(23,40,62,41)(30,44,61,36)(45,52,67,51)(54,66,55,68);(2,4)(5,8,11,31)(6,15,18,27)(7,21,24,34)(9,29,35,63)(10,30,39,61)(12,42,13,32)(14,45,47,67)(16,49,53,20)(17,50,57,46)(19,52,43,51)(22,40,60,41)(23,36,62,44)(25,38,26,28)(33,48,65,58)(37,68,59,66)(54,56,55,64);(5,9)(6,16)(7,22)(8,29)(10,38)(11,35)(12,36)(13,44)(14,46)(15,49)(17,45)(18,53)(19,54)(20,27)(21,40)(23,32)(24,60)(25,61)(26,30)(28,39)(31,63)(33,37)(34,41)(42,62)(43,55)(47,50)(48,68)(51,64)(52,56)(57,67)(58,66)(59,65)

Polycyclic presentation:

2^9|b^2=i,c^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, 406971);