#148720

order	512
nilpotency class	4
isoclinism	isoclinic class of #13371
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#476
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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=di,c^2=d,d^2=i,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,c^b=cf,e^c=eh,g^c=gi,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, 55916);