#136650

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	C₂²×D₄	M₁₆
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

363 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,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, 43846);