#9713324

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

4485 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,d^2=h,e^2=g,f^2=g,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, 9620520);