Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2021 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#11070 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | #120 |
| commutator | C24 | C22×C4×C5 |
| Frattini, cosocle | C25 | C23×C5 |
| prime radical | C25×C5 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C25×C5 ◁ G | |
1202 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)(7,9)(10,11,12,13)(14,15)(16,19)(17,20)(18,21)(22,27)(23,26)(24,28)(25,29);(6,7,8,9)(14,16)(15,19)(17,22)(18,23)(20,27)(21,26)(24,29)(25,28);(14,17,18,24)(15,20,21,28)(16,22,23,29)(19,25,26,27)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4058);