Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10304 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#10675 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C23×C5 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C42×C5 ◁ G | |
426 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5)(6,7,9,11)(8,12,13,10)(14,15,19,21,18,16,17,20);(6,8,9,13)(7,10,11,12)(14,16,18,15)(17,20,19,21)(22,23)(24,26)(25,28)(27,29);(14,15,18,16)(17,21,19,20)(23,26)(24,29)(27,28)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3663);