#9222156

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

4627 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=ghi,d^2=g,f^2=gh,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, 9129352);