Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#24884 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C22×C5×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C4×C5×D4 ◁ G | |
972 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)(12,14)(13,15)(16,17)(20,27)(24,32)(28,33)(29,30);(7,9)(10,12,13,16)(11,14,15,17)(18,19)(20,28)(21,25)(22,26)(23,31)(24,30)(27,33)(29,32);(11,15)(14,17)(18,20,22,27)(19,24,26,32)(21,30,23,29)(25,33,31,28);(7,9)(18,21,22,23)(19,25,26,31)(20,29,27,30)(24,28,32,33)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17872);