Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#9038 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #120 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #386 |
| prime radical | C5×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
468 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7)(8,11)(9,16,17,12)(10,13)(14,18,20,21)(15,19)(23,27,30,33)(24,31,32,34);(6,8,10,15)(7,11,13,19)(9,14,17,20)(12,18,16,21)(22,23,28,27,26,30,25,33)(24,29,31,35,32,37,34,36);(6,9)(7,12)(8,14)(10,17)(11,18)(13,16)(15,20)(19,21)(22,24)(23,29)(25,34)(26,32)(27,35)(28,31)(30,37)(33,36)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2026);