Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#23609 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C23×C5 | C22×D4 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ C22×C42×C5 ◁ G | |
1138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(6,7,8,9)(14,15)(16,25)(17,26)(18,22)(19,23)(20,24)(21,28)(27,29);(10,11)(12,13)(14,16,18,26,20,21,19,27)(15,17,23,25,24,29,22,28);(7,9)(11,13)(14,17)(15,21)(16,24)(18,28)(19,25)(20,29)(22,27)(23,26);(14,18,20,19)(15,22,24,23)(16,26,21,27)(17,28,29,25)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16597);