Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#13147 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C2×D8 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C22×C5×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
394 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5)(6,7)(8,9)(10,11,13,12)(15,21)(16,24)(17,18)(19,26)(22,29)(23,25)(27,28);(10,12,13,11)(14,15,18,22,19,23,27,16,20,24,28,25,26,29,17,21);(7,9)(14,16,17,23,26,22,28,15,20,21,27,29,19,25,18,24)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6135);