#147262

order	512
nilpotency class	4
isoclinism	isoclinic class of #2192
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1879
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

518 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=hi,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,h^a=hi,d^b=dh,e^b=ehi,h^b=hi,d^c=dhi,e^c=ehi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54458);