#439074

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1405 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:

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

Polycyclic presentation:

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

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, 346270);