#146412

order	512
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₈×D₄
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

916 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)(3,4)(5,6)(7,8)(9,10,14,21,16,11,18,25,17,23,15,22,13,20,31,26)(12,24,28,38,27,35,40,37,29,36,32,39,30,19,33,34);(2,4)(9,11,15,26,16,23,31,21,17,20,14,25,13,10,18,22)(12,24,28,38,27,35,40,37,29,36,32,39,30,19,33,34);(6,8)(9,12)(10,19)(11,24)(13,30)(14,32)(15,28)(16,27)(17,29)(18,33)(20,36)(21,37)(22,34)(23,35)(25,39)(26,38)(31,40)

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=i,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,e^c=ei,f^c=fi

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53608);