#148724

order	512
nilpotency class	4
isoclinism	isoclinic class of #13331
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#459
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

566 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(5,6,7,8)(9,10)(11,19)(12,24)(13,29)(14,31)(15,20)(16,22)(17,26)(18,30)(21,35)(23,37)(25,39)(27,36)(28,40)(32,34)(33,38);(1,2)(3,4)(9,11,13,22,15,23,30,14)(10,16,18,19,20,31,29,37)(12,21,25,38,28,26,27,32)(17,24,33,36,35,40,34,39);(2,4)(9,12,13,25,15,28,30,27)(10,17,18,33,20,35,29,34)(11,21,22,38,23,26,14,32)(16,24,19,36,31,40,37,39)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=i,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,c^b=cf,e^c=ehi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55920);