#136712

order	512
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1869
prime radical	#532	C₂×C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

339 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=g,d^2=g,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^b=dg,e^b=egi,f^b=fg,d^c=dgi,e^c=egi,f^c=fgi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43908);