#9712162

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

4531 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=hi,d^2=gh,e^2=g,f^2=ghi,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, 9619358);