#9713336

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

4588 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)(3,7)(4,6)(5,8)(9,10,14,18)(11,15,21,24)(12,16,20,23)(13,17,19,22)(26,29)(31,33)(32,37)(38,39);(1,2,4,6)(3,5,8,7)(9,11)(10,15)(12,19)(13,20)(14,21)(16,22)(17,23)(18,24)(28,36)(32,37)(34,40)(38,39);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,13)(14,20)(15,17)(18,23)(19,21)(22,24)(25,26,30,29)(27,33,35,31)(28,32,36,37)(34,39,40,38);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,27,30,35)(26,31,29,33)(28,34,36,40)(32,38,37,39);(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,24)(18,23)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40);(9,13,14,19)(10,17,18,22)(11,20,21,12)(15,23,24,16)(25,29,30,26)(27,33,35,31)(28,37,36,32)(34,39,40,38)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,d^2=h,f^2=ghi,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, 9620532);