#504235

order	512
nilpotency class	4
isoclinism	isoclinic class of #504125
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 ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

424 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fh,d^2=ei,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,f^c=fi,g^c=gh,h^c=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, 411431);