#498868

order	512
nilpotency class	4
isoclinism	isoclinic class of #498821
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60643	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1538 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:

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

Polycyclic presentation:

2^9|a^2=i,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gi,h^a=hi,c^b=cg,d^b=di,e^b=eh,f^b=fh,g^b=gi,h^b=hi,e^c=ehi,f^c=fhi,g^c=gi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406064);