#466841

order	512
nilpotency class	3
isoclinism	isoclinic class of #457102
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1058 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=ef,d^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^c=de,g^c=gi,h^d=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, 374037);