#495729

order	512
nilpotency class	4
isoclinism	isoclinic class of #483882
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3050	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1018 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)(5,6,11,17,12,19,35,18)(7,20,24,47,25,49,34,44)(8,14,28,41,29,43,61,42)(9,15,10,32,31,16,33,46)(13,22,37,23,39,55,38,56)(21,48,52,66,53,62,60,68)(26,40,58,67,30,45,27,59)(36,54,63,51,65,50,64,57);(5,7,12,25)(6,13,19,39)(8,21,29,53)(9,23,31,56)(10,22,33,55)(11,34,35,24)(14,36,43,65)(15,44,16,47)(17,38,18,37)(20,32,49,46)(26,57,30,51)(27,54,58,50)(28,60,61,52)(40,66,45,68)(41,64,42,63)(48,59,62,67);(2,4)(5,8)(6,14)(7,21)(9,30)(10,27)(11,28)(12,29)(13,36)(15,45)(16,40)(17,41)(18,42)(19,43)(20,48)(22,54)(23,51)(24,52)(25,53)(26,31)(32,59)(33,58)(34,60)(35,61)(37,63)(38,64)(39,65)(44,68)(46,67)(47,66)(49,62)(50,55)(56,57);(5,9,11,10,12,31,35,33)(6,15,17,32,19,16,18,46)(7,22,24,23,25,55,34,56)(8,26,28,58,29,30,61,27)(13,20,37,47,39,49,38,44)(14,40,41,67,43,45,42,59)(21,50,52,57,53,54,60,51)(36,62,63,68,65,48,64,66)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=g,e^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=dh,e^b=ehi,g^b=gi,h^b=hi,d^c=di,e^d=ei

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, 402925);