#502120

order	512
nilpotency class	4
isoclinism	isoclinic class of #500834
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#24266	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

787 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^c=fh,g^c=gh,h^c=hi

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, 409316);