#431851

order	512
nilpotency class	3
isoclinism	isoclinic class of #418518
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

806 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 339047);