Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17386 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#21338 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3470 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(3,4)(5,9)(6,7)(8,10)(11,12,15,19,16,18,23,13)(14,17,22,25,21,24,26,20);(4,7)(8,9)(11,13,15,18,16,19,23,12)(14,20,22,24,21,25,26,17);(3,5,6,10)(4,8,7,9)(11,14)(12,17)(13,20)(15,22)(16,21)(18,24)(19,25)(23,26);(14,21)(17,24)(20,25)(22,26)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279428);