Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5×C8 · Schur multiplier (H2) = C29 |
| direct product | C5×C8×2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C24 |
| commutator | C2 | C24×C5×C8 |
| Frattini, cosocle | C2×C4 | C25×C5 |
| socle | C22×C5 | C24×C4 |
| prime radical | C2×C5×2−1+4 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ G | |
2190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 5 generators on 29 points:
(1,2,3,4,5)(6,7,8,9,10,11,12,13);(15,19)(17,25)(20,22)(23,29);(14,15,18,19)(16,22,24,20)(17,21,25,26)(23,28,29,27);(14,16,18,24)(15,20,19,22)(17,23,25,29)(21,27,26,28);(14,17)(15,21)(16,23)(18,25)(19,26)(20,27)(22,28)(24,29)680 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53317);