#9684206

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

4439 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=ghi,d^2=g,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,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, 9591402);