#439034

order	512
nilpotency class	3
isoclinism	isoclinic class of #418438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1432 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=fi,d^2=ei,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,h^b=hi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 346230);