Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#25912 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C24×C5 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C22×C42×C5 ◁ G | |
1078 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)(10,11,14,17)(12,16,13,15)(18,19,21,23)(20,26,28,33)(22,30,27,25)(24,32,31,29);(6,7)(8,9)(10,12,14,13)(11,15,17,16)(19,23)(24,31)(25,30)(26,33);(7,9)(10,12,14,13)(11,16,17,15)(18,20,27,32,21,28,22,29)(19,24,30,33,23,31,25,26);(10,13,14,12)(11,16,17,15)(19,23)(20,28)(25,30)(29,32)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 18900);