#500845

order	512
nilpotency class	4
isoclinism	isoclinic class of #500843
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

949 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:

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

Polycyclic presentation:

2^9|b^2=ehi,d^2=eh,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,h^a=hi,c^b=cg,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408041);