Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11115 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#11265 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #388 |
| 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)(8,18)(9,23,27,31)(10,16)(11,15)(12,17)(13,20)(14,25,32,26)(19,34,36,28)(21,29)(22,33)(24,37,35,30);(6,8,10,21,12,22,11,20)(7,13,16,18,17,29,15,33)(9,24,25,36,27,35,26,19)(14,30,23,34,32,37,31,28);(6,9,10,25,12,27,11,26)(7,14,15,31,17,32,16,23)(8,19,20,35,22,36,21,24)(13,28,18,30,29,34,33,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4253);