#9726966

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

4738 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,e^2=gi,f^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9634162);