#9706973

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

4200 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gi,d^2=h,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,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, 9614169);