Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#11119 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #394 |
| 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 | |
864 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)(8,18)(9,22,25,14)(10,17)(11,15)(12,16)(13,20)(19,34,36,30)(21,32)(23,37,35,28)(24,31,27,33)(26,29);(6,8,10,20,12,21,11,26)(7,13,15,18,16,29,17,32)(9,23,24,19,25,35,27,36)(14,30,31,28,22,34,33,37);(6,9)(7,14)(8,19)(10,24)(11,27)(12,25)(13,28)(15,31)(16,22)(17,33)(18,34)(20,35)(21,36)(23,26)(29,37)(30,32)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4107);