#495707

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

1042 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)(9,10)(11,20)(12,17)(13,26)(14,24)(15,19)(16,22)(18,28)(21,37)(23,38)(25,35)(27,31)(29,36)(30,34)(32,40)(33,39);(9,11,15,23)(10,16,19,33)(12,21,14,30)(13,27,29,40)(17,31,24,32)(18,34,35,37)(20,36,38,26)(22,25,39,28);(2,4)(9,10)(11,16)(12,24)(13,18)(14,17)(15,19)(20,22)(21,32)(23,33)(25,36)(26,28)(27,34)(29,35)(30,31)(37,40)(38,39);(5,7,8,6)(9,12,15,14)(10,17,19,24)(11,21,23,30)(13,25,29,28)(16,31,33,32)(18,26,35,36)(20,37,38,34)(22,27,39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=gi,e^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=eh,h^b=hi,d^c=di

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, 402903);