Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8304 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#9881 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C2×C22⋊C4 |
| commutator | C24 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| prime radical | C24×C5 | C22×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C22×C42×C5 ◁ G | |
556 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5)(6,7)(8,12)(9,10)(11,13)(14,15)(16,18,20,21)(17,19)(22,23,27,28,26,29,25,24);(7,10)(11,12)(14,16)(15,18,19,21)(17,20)(22,24,25,29,26,28,27,23);(6,8,9,13)(7,11,10,12)(22,25,26,27)(23,28,29,24)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2869);