Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C5 · Schur multiplier (H2) = C29 |
| direct product | C5×#34312 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C25 |
| commutator | C22 | C24×C4×C5 |
| Frattini, cosocle | C23 | C25×C5 |
| prime radical | C2×C5×#548 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
2960 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 5 generators on 29 points:
(1,2,3,4,5)(6,7,9,8)(11,13)(14,15,20,18)(16,23,17,21)(19,26,27,22)(24,28,25,29);(14,16,20,17)(15,21,18,23)(19,24,27,25)(22,28,26,29);(10,11,12,13)(14,17,20,16)(15,21,18,23)(19,25,27,24)(22,28,26,29);(14,18,20,15)(16,21,17,23)(19,26,27,22)(24,28,25,29);(6,8,9,7)(14,19,20,27)(15,22,18,26)(16,24,17,25)(21,28,23,29)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 27300);