Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9239 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#9241 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #386 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| prime radical | C2×C5×#449 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
712 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5)(6,7)(8,10,12,13)(9,11)(14,15,20,23)(16,19,22,18)(17,24)(21,25)(26,28)(27,29);(6,8,9,12)(7,10)(11,13)(14,16,18,15,20,22,19,23)(17,24,26,21,27,29,25,28);(14,17)(15,21)(16,24)(18,26)(19,25)(20,27)(22,29)(23,28)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2229);