Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#20930 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C5×#117 ◁ G | |
1738 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)(7,9)(10,11)(12,13)(14,15,19,23,20,17,18,22)(16,24,28,21,26,29,27,25);(6,7,8,9)(14,16)(15,21)(17,25)(18,28)(19,27)(20,26)(22,29)(23,24);(11,13)(14,17,20,15)(16,25,26,21)(18,22,19,23)(24,28,29,27);(16,26)(21,25)(24,29)(27,28)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13918);