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) = C24 |
| direct product | C5×#11273 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #394 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C2×C5 | #1442 |
| prime radical | C24×C5 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
576 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)(8,12,17,27)(9,19)(10,23)(11,14)(13,21)(15,24)(16,30,20,35)(18,34,33,26)(22,36)(25,37,28,32)(29,31);(6,8,10,18,11,17,24,33)(7,12,15,26,14,27,23,34)(9,20,21,37,22,16,31,32)(13,28,19,35,29,25,36,30);(6,9)(7,13,14,29)(8,16,17,20)(10,21)(11,22)(12,25)(15,19,23,36)(18,32,33,37)(24,31)(26,30)(27,28)(34,35)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4261);