#430634

order	512
nilpotency class	3
isoclinism	isoclinic class of #418484
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×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1113 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=e,d^2=ef,e^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^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, 337830);