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) = C4 |
| direct product | C5×#13117 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #476 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1920 |
| prime radical | C23×C5 | #137 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C2×C4×C5×C8 ◁ G | |
254 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)(6,7,9,13,12,17,11,15)(8,18,21,32,22,35,19,34)(10,24,23,33,27,31,26,16)(14,25,29,20,30,36,28,37);(7,13,17,15)(14,28,30,29)(16,24,33,31)(18,34,35,32);(6,8,9,19,12,22,11,21)(7,14,15,29,17,30,13,28)(10,25,23,37,27,36,26,20)(16,32,31,35,33,34,24,18)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6105);