#464255

order	512
nilpotency class	3
isoclinism	isoclinic class of #457096
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1743 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 371451);