#9222113

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

4453 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=i,d^2=g,e^2=g,f^2=h,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, 9129309);