Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8479 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#8487 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #379 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| prime radical | C5×#3534 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
832 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)(8,9,12,13)(10,11)(14,15,18,22,19,23,17,27)(16,24,25,20,26,21,28,29);(6,8)(7,9)(10,12)(11,13)(14,16)(15,20)(17,28)(18,25)(19,26)(21,22)(23,29)(24,27);(6,9)(7,8)(10,13)(11,12)(14,16)(15,21)(17,25)(18,28)(19,26)(20,22)(23,24)(27,29)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1475);