#399290

order	512
nilpotency class	3
isoclinism	isoclinic class of #395252
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

939 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,21)(13,27)(15,19)(16,20)(17,25)(18,30)(22,29)(23,36)(24,33)(26,32)(28,34)(31,37)(35,39)(38,40);(2,5)(6,8)(9,11,15,20,16,19,14,10)(12,17,25,21,26,33,24,32)(13,22,29,27,30,34,28,18)(23,31,38,35,39,40,37,36);(9,12,15,25,16,26,14,24)(10,17,11,21,20,33,19,32)(13,23,29,38,30,39,28,37)(18,31,22,35,27,40,34,36);(1,3,4,7)(2,6,5,8)(9,13,14,28,16,30,15,29)(10,18,19,34,20,27,11,22)(12,23,24,37,26,39,25,38)(17,31,32,36,33,40,21,35)

Polycyclic presentation:

2^9|b^2=f,c^2=f,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,g^c=gi,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, 306486);