#148984

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

528 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:

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

Polycyclic presentation:

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

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, 56180);