#9161727

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

4559 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)(11,18)(12,15)(13,17)(14,16)(19,24)(20,22)(21,23)(25,26,31,35)(27,36,38,32)(28,33,37,30)(29,34,39,40);(1,2)(3,8)(4,6)(5,7)(9,11,15,14)(10,16,12,18)(13,21,22,19)(17,24,20,23)(25,27,31,38)(26,32,35,36)(28,29,37,39)(30,34,33,40);(1,3)(2,4)(5,6)(7,8)(10,12)(16,18)(17,20)(23,24)(25,28,31,37)(26,33,35,30)(27,29,38,39)(32,40,36,34);(3,6)(4,8)(9,12,15,10)(11,18,14,16)(13,17,22,20)(19,23,21,24)(25,28)(26,30)(27,29)(31,37)(32,34)(33,35)(36,40)(38,39);(1,4,5,8)(2,6,7,3)(9,13,15,22)(10,17,12,20)(11,19,14,21)(16,23,18,24)(25,29)(26,34)(27,37)(28,38)(30,36)(31,39)(32,33)(35,40);(1,2)(3,4)(5,7)(6,8)(9,14,15,11)(10,18,12,16)(13,19,22,21)(17,23,20,24)(25,30)(26,28)(27,34)(29,32)(31,33)(35,37)(36,39)(38,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=h,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9068923);