#509156

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

775 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)(5,6,9,15,30,44,56,34,11,17,31,33,10,16,28,12)(7,13,20,35,18,37,46,60,22,39,52,59,21,38,51,32)(8,23,24,40,25,54,57,42,27,55,58,41,26,43,29,14)(19,47,49,64,53,68,67,63,50,66,65,62,45,61,48,36);(6,12)(7,18)(8,24)(9,28)(10,30)(13,35)(14,40)(15,16)(17,34)(19,48)(21,22)(25,29)(26,57)(27,58)(31,56)(32,37)(33,44)(38,60)(39,59)(41,42)(43,54)(45,49)(46,51)(47,61)(50,67)(53,65)(62,64)(66,68);(2,4)(5,7)(6,13)(8,19)(9,20)(10,21)(11,22)(12,32)(14,36)(15,35)(16,38)(17,39)(18,30)(23,47)(24,49)(25,53)(26,45)(27,50)(28,51)(29,48)(31,52)(33,59)(34,60)(37,44)(40,64)(41,62)(42,63)(43,61)(46,56)(54,68)(55,66)(57,67)(58,65);(2,4)(5,8,11,27)(6,14,17,42)(7,19,22,50)(9,29,31,57)(10,25,30,26)(12,23,34,55)(13,36,39,63)(15,43,33,54)(16,40,44,41)(18,45,21,53)(20,48,52,67)(24,56,58,28)(32,47,60,66)(35,61,59,68)(37,62,38,64)(46,65,51,49)

Polycyclic presentation:

2^9|a^2=e,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, 416352);