#9713761

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

4479 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)(4,5)(6,8)(9,10,15,12)(11,18,14,16)(13,17,22,20)(19,24,21,23)(26,29)(31,33)(32,37)(38,39);(3,8)(6,7)(9,11,15,14)(10,16,12,18)(13,21,22,19)(17,24,20,23)(28,36)(32,37)(34,40)(38,39);(1,3)(2,6)(4,7)(5,8)(10,12)(16,18)(17,20)(23,24)(25,26,30,29)(27,33,35,31)(28,32,36,37)(34,39,40,38);(1,2,5,4)(3,7,8,6)(9,12,15,10)(11,18,14,16)(13,20,22,17)(19,24,21,23)(25,27,30,35)(26,31,29,33)(28,34,36,40)(32,38,37,39);(1,3)(2,7)(4,6)(5,8)(9,13,15,22)(10,17,12,20)(11,19,14,21)(16,23,18,24)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40);(1,4,5,2)(3,7,8,6)(9,14,15,11)(10,18,12,16)(13,19,22,21)(17,23,20,24)(25,29,30,26)(27,33,35,31)(28,37,36,32)(34,39,40,38)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,d^2=h,e^2=g,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, 9620957);