Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17411 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#22478 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C24 | C24×C5 |
| prime radical | C22×C5×D4 | C23 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C5×C4⋊C4 ◁ G | |
1132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(7,10)(11,13)(14,15)(16,20)(17,25,27,21)(18,22)(19,23)(24,29,26,28);(6,7)(8,11)(9,10)(12,13)(14,16,19,18)(15,20,23,22)(17,24,27,26)(21,28,25,29);(6,8,9,12)(7,11,10,13)(14,17)(15,21)(16,24)(18,26)(19,27)(20,28)(22,29)(23,25);(7,10)(11,13)(14,18,19,16)(15,22,23,20)(17,26,27,24)(21,29,25,28)275 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15466);