Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#11003 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #392 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C2×C5 | #1438 |
| prime radical | C5×#3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
1292 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)(8,17)(9,21,24,32)(10,25)(11,15)(12,16)(13,19)(14,23,31,22)(18,30,35,27)(20,28)(26,29)(33,34,37,36);(6,8)(7,13)(9,18)(10,26)(11,19)(12,20)(14,30)(15,28)(16,29)(17,25)(21,37)(22,36)(23,34)(24,35)(27,31)(32,33);(6,9)(7,14)(8,18)(10,23)(11,22)(12,24)(13,27)(15,32)(16,31)(17,33)(19,36)(20,35)(21,25)(26,34)(28,37)(29,30)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3991);