#9781027

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

4297 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9688223);