Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7694 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C4 |
| direct product | C5×#7717 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C43×C5 ◁ G | |
352 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5)(6,7,9,11,10,8,12,13)(14,15)(16,22,24,19)(17,25)(18,21)(20,26)(23,27,29,28)(30,31)(32,38,40,45)(33,37)(34,36)(35,39,43,41)(42,44);(6,8)(7,10)(9,11)(12,13)(14,16,18,24)(15,19,21,22)(17,23,26,29)(20,27,25,28)(30,32)(31,35)(33,41)(34,40)(36,43)(37,45)(38,44)(39,42);(7,8)(11,13)(14,17,18,26)(15,20,21,25)(16,23,24,29)(19,27,22,28)(31,36)(35,43)(37,44)(38,45)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 705);