#9221214

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

4524 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9128410);