#111147

order	512
nilpotency class	3
isoclinism	isoclinic class of #9183
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1437
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

651 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)(3,8)(4,6)(5,7)(9,10,14,19,15,20,13,18)(11,21,23,30,24,34,28,16)(12,17,25,31,26,32,27,33)(22,36,39,29,38,35,37,40);(1,3,4,2,5,6,8,7)(9,11)(10,16,20,30)(12,22)(13,23)(14,28)(15,24)(17,29,32,40)(18,21,19,34)(25,37)(26,38)(27,39)(31,36,33,35);(9,12,14,25,15,26,13,27)(10,17,19,31,20,32,18,33)(11,22,28,37,24,38,23,39)(16,29,34,36,30,40,21,35)

Polycyclic presentation:

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

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, 18343);