Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C43 | C23 |
| socle, prime radical | C23 | #385 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
392 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,7)(3,5,8,6)(10,15)(11,16)(12,13)(17,18,21,19,20,22,24,23);(9,10)(11,12)(13,15)(14,16)(17,19,21,22,20,23,24,18);(2,5)(3,8)(6,7)(9,11,14,15)(10,12,16,13)Polycyclic presentation:
2^9|a^2=f,b^2=fgh,c^2=g,d^2=g,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dg,e^b=eg,d^c=dg,e^c=eh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6539);