#494287

order	512
nilpotency class	4
isoclinism	isoclinic class of #33522
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1062 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)(5,6,8,7)(10,15)(11,19)(12,22)(13,25)(17,30)(18,34)(20,38)(21,37)(23,27)(24,28)(26,36)(29,32)(33,35)(39,40);(2,4)(5,7,8,6)(9,10)(11,16)(12,23)(13,27)(14,17)(15,22)(18,25)(19,29)(20,35)(21,31)(24,32)(26,34)(28,33)(30,36)(37,39)(38,40);(9,11,14,21)(10,16,17,31)(12,24,26,38)(13,28,25,20)(15,29,30,39)(18,35,27,33)(19,36,37,22)(23,32,34,40);(10,17)(15,30)(16,31)(18,27)(23,34)(29,39)(32,40)(33,35)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,e^2=hi,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,h^a=hi,d^b=di,e^b=eh,f^b=fh,h^b=hi,e^c=eh,f^c=fh,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, 401483);