#507453

order	512
nilpotency class	4
isoclinism	isoclinic class of #501764
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#476
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

408 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)(3,7)(4,5)(6,8)(9,10,15,20,13,19,16,31)(11,21,24,37,23,17,14,29)(12,25,28,32,27,36,30,40)(18,22,34,26,33,38,35,39);(2,5)(6,7)(9,11,15,24,13,23,16,14)(10,17,20,29,19,21,31,37)(12,26,28,38,27,39,30,22)(18,32,34,36,33,40,35,25);(9,12,14,26,16,28,23,38,13,27,24,39,15,30,11,22)(10,18,17,25,20,35,29,40,19,33,21,36,31,34,37,32);(1,3,4,8)(2,6,5,7)(10,19)(17,21)(18,33)(20,31)(25,36)(29,37)(32,40)(34,35)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414649);