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) = C26 |
| direct product | C5×#11033 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #394 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C2×C5 | #1438 |
| prime radical | C2×C5×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
1264 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)(8,16,19,12)(9,20,22,13)(10,24)(11,14)(15,27)(17,28)(18,29,25,32)(21,37)(23,31,26,33)(30,36)(34,35);(6,8)(7,12)(9,21)(10,25)(11,19)(13,30)(14,16)(15,32)(17,23)(18,27)(20,37)(22,36)(24,29)(26,35)(28,31)(33,34);(6,9)(7,13)(8,17)(10,26)(11,22)(12,28)(14,20)(15,33)(16,34)(18,36)(19,35)(21,25)(23,27)(24,31)(29,37)(30,32)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4021);