Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#24207 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C22×C5×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C4×C5×Q8 ◁ G | |
1076 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)(6,7)(8,9)(10,11)(12,16)(13,14)(15,17)(18,19)(20,22)(21,28)(23,26)(24,27)(25,29)(30,33)(31,32);(11,14)(15,16)(18,20,24,26)(19,22,27,23)(21,25,31,33)(28,29,32,30);(7,9)(18,21,24,31)(19,25,27,33)(20,28,26,32)(22,30,23,29);(10,12,13,17)(11,15,14,16)(18,22,24,23)(19,20,27,26)(21,29,31,30)(25,32,33,28)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17195);