#9713333

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

4243 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,d^2=h,e^2=g,f^2=gi,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, 9620529);