#507511

order	512
nilpotency class	4
isoclinism	isoclinic class of #501766
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

438 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=eg,c^2=ef,d^2=e,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,d^b=di,g^c=gi

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, 414707);