#9778469

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

4616 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)(11,20)(12,22)(13,17)(14,21)(15,16)(18,23)(19,24)(26,29)(31,33)(32,37)(38,39);(1,2,7,8)(3,4,6,5)(10,16)(11,13)(18,22)(19,21)(28,36)(32,37)(34,40)(38,39);(1,3,7,6)(2,5,8,4)(9,11,15,13)(10,17,16,20)(12,19,23,21)(14,18,24,22)(25,26,30,29)(27,33,35,31)(28,32,36,37)(34,39,40,38);(1,4,7,5)(2,3,8,6)(9,12,15,23)(10,18,16,22)(11,21,13,19)(14,20,24,17)(25,27,30,35)(26,31,29,33)(28,34,36,40)(32,38,37,39);(1,5,7,4)(2,3,8,6)(9,13,15,11)(10,17,16,20)(12,19,23,21)(14,22,24,18)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40);(1,6,7,3)(2,5,8,4)(9,14)(10,19)(11,18)(12,20)(13,22)(15,24)(16,21)(17,23)(25,29,30,26)(27,33,35,31)(28,37,36,32)(34,39,40,38)

Polycyclic presentation:

2^9|b^2=gi,c^2=ghi,d^2=ghi,e^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dghi,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9685665);