#507428

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	#3050	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

528 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)(9,10,15,19,13,18,22,20)(11,21,14,32,24,16,31,33)(12,25,29,35,27,39,30,40)(17,34,37,28,36,23,38,26);(1,3,4,8)(2,5,6,7)(9,11,22,31,13,24,15,14)(10,16,20,32,18,21,19,33)(12,26,30,23,27,28,29,34)(17,35,38,25,36,40,37,39);(9,12,14,28,15,30,24,34,13,27,31,26,22,29,11,23)(10,17,16,25,20,37,32,35,18,36,21,39,19,38,33,40);(2,6)(5,7)(10,18)(16,21)(17,36)(19,20)(25,39)(32,33)(35,40)(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,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, 414624);