Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#23666 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C23×C5 | C22×D4 |
| prime radical | C24×C5 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ C22×C42×C5 ◁ G | |
910 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 33 points:
(1,2,3,4,5)(10,11)(12,17)(13,15)(14,16)(18,19,22,26,23,27,21,25)(20,28,29,32,30,33,31,24);(6,7)(8,9)(10,12,14,13)(11,15,16,17);(7,9)(10,13,14,12)(11,15,16,17)(18,20)(19,24)(21,29)(22,31)(23,30)(25,28)(26,33)(27,32);(18,21,23,22)(19,25,27,26)(20,29,30,31)(24,28,32,33)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16654);