Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18517 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| socle | C22×C5 | C2×#120 |
| prime radical | C5×D4×#41 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ G | |
1204 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)(6,7,9,12)(8,10,13,11)(14,15,19,24)(16,25,28,29)(17,22,18,23)(20,26,21,27);(7,10)(8,13)(11,12)(14,16)(15,20)(17,26)(18,27)(19,28)(21,24)(22,25)(23,29);(14,16)(15,21)(17,26)(18,27)(19,28)(20,24)(22,29)(23,25);(14,17,19,18)(15,22,24,23)(16,26,28,27)(20,25,21,29)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11505);