#483560

order	512
nilpotency class	3
isoclinism	isoclinic class of #3003
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	#532	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

470 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=f,e^2=g,f^2=h,h^2=i,b^a=be,c^a=cg,e^a=eg,d^b=di,e^b=eg

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390756);