#134356

order	512
nilpotency class	3
isoclinism	isoclinic class of #8674
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#380
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1797
prime radical	C₂²×D₄	M₁₆
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

414 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)(3,5)(4,7)(6,8)(9,10,13,19,15,18,28,33,14,17,27,32,12,25,26,38)(11,20,22,35,24,30,37,39,23,34,36,40,21,16,29,31);(9,11,12,21,14,23,15,24)(10,16,18,20,17,30,25,34)(13,22,26,29,27,36,28,37)(19,31,33,35,32,39,38,40);(2,5)(3,8)(6,7)(10,17)(13,27)(16,30)(18,25)(20,34)(22,36)(26,28)(29,37)

Polycyclic presentation:

2^9|a^2=f,b^2=d,d^2=i,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,e^c=egi,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41552);