Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11000 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#11109 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #390 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C2×C5 | #1439 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
852 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)(9,21,23,31)(10,24)(11,15)(12,16)(13,19)(14,22,32,26)(18,30,36,27)(20,28)(25,29)(33,35,37,34);(6,8,10,19,12,20,11,25)(7,13,15,17,16,29,24,28)(9,18,22,34,23,36,26,35)(14,30,31,37,32,27,21,33);(6,9)(7,14)(8,18)(10,22)(11,26)(12,23)(13,27)(15,31)(16,32)(17,33)(19,34)(20,36)(21,24)(25,35)(28,37)(29,30)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4097);