#502111

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
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	C₂²×D₄	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

770 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)(9,10,14,20,33,24,17,26,13,19,15,21,16,11,22,25)(12,27,30,23,36,40,35,37,29,38,31,39,32,28,34,18);(2,4)(5,6)(7,8)(9,11,14,25,33,10,17,20,13,24,15,26,16,19,22,21)(12,28,30,18,36,27,35,23,29,40,31,37,32,38,34,39);(6,8)(9,12,13,29)(10,18,19,37)(11,23,24,39)(14,34,15,35)(16,36,33,32)(17,31,22,30)(20,28,21,40)(25,27,26,38);(2,4)(10,19)(11,24)(18,37)(20,21)(23,39)(25,26)(27,38)(28,40)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,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, 409307);