#9789069

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

4248 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)(11,16)(12,18)(13,19)(14,21)(15,17)(20,24)(22,23)(25,26,31,33)(27,35,29,32)(28,38,34,30)(36,37,39,40);(1,2,4,6)(3,5,8,7)(9,11,15,12)(10,16,17,18)(13,21,23,20)(14,22,24,19)(25,27,31,29)(26,32,33,35)(28,36,34,39)(30,37,38,40);(1,2)(3,5)(4,6)(7,8)(10,17)(11,12)(19,22)(20,21)(26,33)(28,34)(32,35)(36,39);(9,12,15,11)(10,18,17,16)(13,20,23,21)(14,19,24,22)(25,28,31,34)(26,30,33,38)(27,36,29,39)(32,37,35,40);(1,2)(3,7)(4,6)(5,8)(9,13)(10,19)(11,21)(12,20)(14,16)(15,23)(17,22)(18,24)(25,29,31,27)(26,32,33,35)(28,39,34,36)(30,37,38,40);(1,3)(2,5)(4,8)(6,7)(9,14,15,24)(10,20,17,21)(11,22,12,19)(13,18,23,16)(25,30)(26,34)(27,37)(28,33)(29,40)(31,38)(32,39)(35,36)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9696265);