#494510

order	512
nilpotency class	4
isoclinism	isoclinic class of #33522
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#508	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1354 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401706);