#512233

order	512
nilpotency class	4
isoclinism	isoclinic class of #511893
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

481 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=fh,c^2=eh,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,g^a=gh,h^a=hi,g^b=gi,g^c=gi,e^d=ei,f^d=fi,g^d=ghi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419429);