#509162

order	512
nilpotency class	4
isoclinism	isoclinic class of #509155
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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 ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

774 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)(5,6,14,41,11,18,38,55,13,20,10,17,12,19,37,54)(7,21,56,48,25,58,35,15,27,59,24,47,26,52,42,49)(8,16,43,33,39,34,53,44,31,9,29,51,30,40,67,36)(22,23,46,50,63,64,65,45,57,62,60,32,28,61,68,66);(5,7,13,27)(6,15,20,49)(8,28,31,63)(9,32,16,50)(10,35,14,42)(11,26,12,25)(17,52,41,58)(18,48,19,47)(21,54,59,55)(22,39,57,30)(23,51,62,33)(24,37,56,38)(29,46,43,60)(34,66,40,45)(36,64,44,61)(53,65,67,68);(2,4)(5,8)(6,16)(7,22)(9,20)(10,29)(11,39)(12,30)(13,31)(14,43)(15,45)(17,51)(18,34)(19,40)(21,23)(24,60)(25,63)(26,28)(27,57)(32,47)(33,41)(35,65)(36,54)(37,67)(38,53)(42,68)(44,55)(46,56)(48,50)(49,66)(52,61)(58,64)(59,62);(2,4)(5,9,13,16)(6,8,20,31)(7,23,27,62)(10,36,14,44)(11,34,12,40)(15,46,49,60)(17,53,41,67)(18,30,19,39)(21,57,59,22)(24,45,56,66)(25,61,26,64)(28,58,63,52)(29,54,43,55)(32,42,50,35)(33,37,51,38)(47,68,48,65)

Polycyclic presentation:

2^9|a^2=ei,b^2=i,d^2=i,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,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, 416358);