#9160913

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

4426 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)(11,19)(12,22)(13,18)(14,16)(15,17)(20,24)(21,23)(25,26)(27,31)(28,32)(29,40)(30,34)(33,37)(35,36)(38,39);(1,2)(3,8)(4,6)(5,7)(9,11,15,14)(10,16,17,19)(12,20,18,23)(13,24,22,21)(25,27,30,36)(26,31,34,35)(28,37,39,29)(32,33,38,40);(1,3)(2,4)(5,6)(7,8)(10,17)(12,18)(16,19)(20,23)(25,26,30,34)(27,35,36,31)(28,32,39,38)(29,33,37,40);(3,6)(4,8)(9,12)(10,13)(11,20)(14,23)(15,18)(16,24)(17,22)(19,21)(25,26)(27,31)(28,38)(29,33)(30,34)(32,39)(35,36)(37,40);(1,4,5,8)(2,6,7,3)(9,13)(10,18)(11,21)(12,17)(14,24)(15,22)(16,20)(19,23)(25,28,30,39)(26,32,34,38)(27,29,36,37)(31,40,35,33);(1,2)(3,4)(5,7)(6,8)(9,14,15,11)(10,19,17,16)(12,23,18,20)(13,21,22,24)(25,29)(26,33)(27,28)(30,37)(31,38)(32,35)(34,40)(36,39)

Polycyclic presentation:

2^9|b^2=h,c^2=gh,e^2=i,f^2=g,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, 9068109);