#9064909

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

4421 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=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, 8972105);