Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C211×C4 |
| direct product | C2×#58251 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7572 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 26 points:
(1,2);(3,4)(5,9)(6,7)(8,10)(12,14)(16,17)(19,20,24,25)(21,23,22,26);(3,5,7,10)(4,8,6,9)(11,12,15,14)(13,17,18,16)(20,25)(21,22);(5,10)(8,9)(19,21,24,22)(20,26,25,23);(3,5)(4,9)(6,8)(7,10)(11,13,15,18)(12,16,14,17)(19,22,24,21)(20,23,25,26);(3,6,7,4)(5,8,10,9)(11,14,15,12)(13,17,18,16)(19,23)(20,21)(22,25)(24,26)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8011723);