#507430

order	512
nilpotency class	4
isoclinism	isoclinic class of #501766
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	#3050	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

509 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)(4,6)(5,8)(9,10,14,19,16,20,15,32)(11,17,24,34,22,33,25,40)(12,26,28,31,29,30,18,13)(21,23,27,37,35,36,39,38);(1,3,4,8)(2,5,6,7)(9,11,14,24,16,22,15,25)(10,17,19,34,20,33,32,40)(12,27,28,35,29,39,18,21)(13,23,26,37,31,36,30,38);(9,12,13,20,15,28,30,32,16,29,31,10,14,18,26,19)(11,21,23,34,25,27,38,33,22,35,36,40,24,39,37,17);(2,6)(5,7)(11,22)(17,33)(21,35)(23,36)(24,25)(27,39)(34,40)(37,38)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,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, 414626);