Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#20455 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator | C4 | C22×C42×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C24×C5 | C2×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C5×#117 ◁ G | |
1126 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)(6,7,9,8)(14,15,17,22,18,21,25,19)(16,20,24,27,23,28,29,26);(10,11,12,13)(15,19)(17,25)(20,26)(21,22)(24,29)(27,28);(6,8,9,7)(14,16)(15,20)(17,24)(18,23)(19,26)(21,28)(22,27)(25,29);(15,21)(16,23)(19,22)(24,29)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13443);