#500706

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3287	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

1109 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407902);