Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#12825 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #476 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1802 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C2×C4×C5×C8 ◁ G | |
400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,10,18,13,19,12,16)(8,20,22,14,23,17,11,28)(9,24,26,35,27,37,25,33)(15,30,31,21,32,29,34,36);(6,8,10,22,13,23,12,11)(7,14,16,20,19,28,18,17)(9,21,25,30,27,36,26,29)(15,24,31,35,32,37,34,33);(6,9)(7,15)(8,21)(10,25)(11,29)(12,26)(13,27)(14,24)(16,31)(17,33)(18,34)(19,32)(20,35)(22,30)(23,36)(28,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5813);