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×#18522 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| socle | C22×C5 | C2×#120 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ G | |
1000 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)(7,11)(8,9)(12,13)(14,15)(16,25)(17,28)(18,22)(19,23)(20,24)(21,27)(26,29);(6,7,10,12)(8,11,9,13)(14,16,20,17)(15,21,24,29)(18,26,19,27)(22,25,23,28);(6,8,10,9)(7,11,12,13)(14,17,20,16)(15,21,24,29)(18,27,19,26)(22,25,23,28);(14,18,20,19)(15,22,24,23)(16,26,17,27)(21,25,29,28)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11510);