Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24404 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#25566 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24×C5 |
| prime radical | C5×#534 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1122 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)(6,7)(8,12)(9,11)(10,13)(14,15)(16,20)(17,21)(18,22)(19,23)(24,25)(26,29)(27,28);(6,8,9,13)(7,10,11,12)(14,16)(15,20)(17,27)(18,25)(19,26)(21,28)(22,24)(23,29);(14,17,19,22)(15,21,23,18)(16,24)(20,28)(25,29)(26,27);(7,11)(10,12)(14,18,19,21)(15,22,23,17)(16,25,26,28)(20,24,29,27)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 18554);