Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#22993 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C24 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C23×C5 | C23×C4 |
| prime radical | C5×#392 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ G | |
998 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)(10,11,13,12)(14,15,19,21,18,22,17,26)(16,20,24,29,25,23,27,28);(11,12)(14,15,17,26,18,22,19,21)(16,23,27,29,25,20,24,28);(7,9)(14,16)(15,20)(17,27)(18,25)(19,24)(21,29)(22,23)(26,28);(10,12,13,11)(14,17,18,19)(15,21,22,26)(16,24,25,27)(20,28,23,29)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15981);