Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#24251 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#2535 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C4×C5×D4 ◁ G | |
1456 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)(7,8)(10,11)(12,13)(14,15,18,22)(16,24,27,20)(17,23,19,21)(25,29,26,28);(6,7,9,8)(14,16)(15,20)(17,26)(18,27)(19,25)(21,28)(22,24)(23,29);(11,13)(15,21)(16,25)(17,19)(22,23)(26,27)(28,29);(6,8,9,7)(14,17,18,19)(15,21,22,23)(16,26,27,25)(20,28,24,29)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17239);