Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#25915 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#2619 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C5×#117 ◁ G | |
1110 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)(19,23)(20,27)(21,28)(24,31)(29,33)(30,32);(6,7)(8,9)(10,12,14,13)(11,15,17,16)(18,19,22,23)(20,24,29,30)(21,26,28,25)(27,32,33,31);(7,9)(10,12,14,13)(11,16,17,15)(18,20,28,33,22,29,21,27)(19,24,25,31,23,30,26,32);(10,13,14,12)(11,16,17,15)(18,21,22,28)(19,25,23,26)(20,27,29,33)(24,31,30,32)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 18903);