#9143419

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

4446 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gi,d^2=g,e^2=gh,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, 9050615);