#8898161

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

4597 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=hi,d^2=g,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, 8805357);