#424284

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1545 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)(9,10,14,19,15,20,29,31)(11,21,23,18,24,35,13,16)(12,17,27,33,28,34,39,40)(22,36,37,32,25,38,26,30);(9,11,15,24)(10,16,20,18)(12,25,28,22)(13,29,23,14)(17,32,34,30)(19,35,31,21)(26,27,37,39)(33,36,40,38);(1,2,3,4)(9,12,14,27,15,28,29,39)(10,17,19,33,20,34,31,40)(11,22,23,37,24,25,13,26)(16,30,21,36,18,32,35,38);(6,8)(9,13,14,11,15,23,29,24)(10,18,19,35,20,16,31,21)(12,26,27,22,28,37,39,25)(17,32,33,38,34,30,40,36)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=ef,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,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, 331480);