#9713451

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

4445 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=gi,d^2=h,e^2=g,f^2=hi,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, 9620647);