#9716206

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

4369 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=g,e^2=h,f^2=h,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, 9623402);