#499773

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

Polycyclic presentation:

2^9|b^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, 406969);