#9206531

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

4347 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)(3,4)(5,6)(7,8)(9,10,14,15)(11,17,18,12)(13,21,23,16)(19,20,24,22)(26,31)(29,32)(33,38)(37,39);(2,6)(3,7)(9,10,14,15)(11,12,18,17)(13,16,23,21)(19,20,24,22)(28,36)(33,38)(35,40)(37,39);(1,3,5,7)(2,4,6,8)(10,15)(11,18)(16,21)(19,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(1,4)(2,7)(3,6)(5,8)(9,11,14,18)(10,12,15,17)(13,19,23,24)(16,20,21,22)(25,27,30,34)(26,32,31,29)(28,35,36,40)(33,39,38,37);(9,12)(10,11)(13,22)(14,17)(15,18)(16,24)(19,21)(20,23)(25,28,30,36)(26,33,31,38)(27,35,34,40)(29,37,32,39);(9,13,14,23)(10,16,15,21)(11,19,18,24)(12,20,17,22)(25,29,30,32)(26,34,31,27)(28,37,36,39)(33,40,38,35)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=g,d^2=h,e^2=ghi,f^2=h,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, 9113727);