Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #43535 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,7)(3,6,8,5)(9,10)(11,18)(12,16)(13,21)(14,15)(17,23)(19,20)(22,24)(25,26,32,29)(27,33,37,31)(28,30,34,39)(35,40,38,36);(3,8)(5,6)(10,15)(12,18)(17,21)(20,22)(25,27,32,37)(26,33,29,31)(28,38,34,35)(30,36,39,40);(1,3)(2,5)(4,8)(6,7)(10,15)(11,16)(17,21)(19,24)(25,28,32,34)(26,30,29,39)(27,35,37,38)(31,36,33,40);(3,8)(5,6)(9,11)(10,12)(13,19)(14,16)(15,18)(17,20)(21,22)(23,24)(25,29,32,26)(27,31,37,33)(28,30,34,39)(35,40,38,36);(1,3)(2,6)(4,8)(5,7)(9,12)(10,16)(11,15)(13,22)(14,18)(17,19)(20,23)(21,24)(25,30)(26,34)(27,36)(28,29)(31,38)(32,39)(33,35)(37,40);(1,2)(3,5)(4,7)(6,8)(9,13,14,23)(10,17,15,21)(11,19,16,24)(12,20,18,22)(25,31)(26,27)(28,36)(29,37)(30,35)(32,33)(34,40)(38,39)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=ghi,d^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9067178);