#8898264

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4456 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=ghi,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8805460);