#498881

order	512
nilpotency class	4
isoclinism	isoclinic class of #498812
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	C₂×#3414	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1524 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)(7,19)(8,25)(9,15)(10,33)(11,17)(12,18)(13,22)(14,29)(16,36)(20,38)(21,47)(23,52)(24,53)(26,63)(27,62)(28,46)(30,48)(31,44)(32,50)(34,40)(35,43)(37,49)(39,58)(41,60)(42,54)(45,65)(51,56)(55,68)(57,67)(59,61)(64,66);(1,2)(3,4)(5,7,12,24)(6,13,18,41)(8,26,31,54)(9,23,11,21)(10,34,37,58)(14,42,46,63)(15,40,17,39)(16,47,50,52)(19,49,53,33)(20,27,57,30)(22,32,60,36)(25,61,44,66)(28,55,29,56)(35,64,65,59)(38,45,67,43)(48,68,62,51);(2,4)(5,8)(6,14)(7,20)(9,27)(10,35)(11,30)(12,31)(13,38)(15,43)(16,48)(17,45)(18,46)(19,51)(21,54)(22,59)(23,26)(24,57)(25,33)(28,32)(29,36)(34,56)(37,65)(39,42)(40,63)(41,67)(44,49)(47,61)(50,62)(52,66)(53,68)(55,58)(60,64);(5,9,12,11)(6,15,18,17)(7,21,24,23)(8,27,31,30)(10,32,37,36)(13,39,41,40)(14,43,46,45)(16,33,50,49)(19,47,53,52)(20,54,57,26)(22,58,60,34)(25,62,44,48)(28,65,29,35)(38,42,67,63)(51,61,68,66)(55,64,56,59)

Polycyclic presentation:

2^9|b^2=i,d^2=i,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,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=fh,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, 406077);