Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#21902 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C22 | C24×C8 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C24 | C23×C4 |
| prime radical | C26 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3594 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)(6,8)(7,10)(11,12)(13,15)(14,16)(17,18)(19,20,21,23,22,24,25,26);(3,5)(4,7)(6,10)(8,9)(11,13,14,17)(12,15,16,18);(5,10)(7,9)(12,16)(15,18);(4,8)(7,9)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281930);