Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18252 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#18259 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1132 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)(10,13)(14,15,20,17)(16,24,27,29)(18,23,19,22)(21,28,25,26);(6,7)(8,10)(9,11)(12,13)(14,16)(15,21)(17,25)(18,26)(19,28)(20,27)(22,24)(23,29);(6,8)(7,10)(9,12)(11,13)(14,17,20,15)(16,25,27,21)(18,22,19,23)(24,28,29,26);(14,18,20,19)(15,22,17,23)(16,26,27,28)(21,24,25,29)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11247);