#9206689

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

4443 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9113885);