#500846

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

Polycyclic presentation:

2^9|b^2=eh,d^2=ehi,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, 408042);