Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17392 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25×C4 |
| direct product | C5×#21747 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C24 | C24×C5 |
| prime radical | C23×C5×D4 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C24×C4×C5 ◁ G | |
1570 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(6,7)(8,12)(9,11)(10,13)(14,15)(16,19)(17,24,25,20)(18,21)(22,28,29,26)(23,27);(6,8)(7,10)(9,13)(11,12)(14,16,18,23)(15,19,21,27)(17,22,25,29)(20,26,24,28);(8,13)(10,12)(14,17)(15,20)(16,22)(18,25)(19,26)(21,24)(23,29)(27,28);(7,11)(10,12)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14735);