Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#17381 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C2×C22⋊C4 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C22×C5 | C22×C22⋊C4 |
| prime radical | C2×C5×#525 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C24×C4×C5 ◁ G | |
1446 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)(8,12,13,10)(9,11)(14,15,18,22,19,23,17,21)(16,20,26,29,24,27,25,28);(6,8)(7,10)(9,13)(11,12)(14,16)(15,20)(17,25)(18,26)(19,24)(21,28)(22,29)(23,27);(16,24)(20,27)(25,26)(28,29);(14,17,19,18)(15,21,23,22)(16,25,24,26)(20,28,27,29)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10369);