Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#11363 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #397 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C2×C5 | #1445 |
| prime radical | C24×C5 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
560 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)(6,7,11,17,12,15,10,16)(8,13,21,29,19,30,20,28)(9,22,24,32,26,33,25,14)(18,31,35,36,34,37,23,27);(6,8,10,21,12,19,11,20)(7,13,17,28,15,30,16,29)(9,23,24,34,26,35,25,18)(14,31,33,36,32,37,22,27);(6,9,10,24,12,26,11,25)(7,14,17,33,15,32,16,22)(8,18,21,23,19,34,20,35)(13,27,28,31,30,36,29,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4351);