Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23924 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#24072 |
| 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 ◁ C2×C5×#117 ◁ G | |
1002 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)(18,19,24,27)(20,25,29,33)(21,30,28,32)(22,26,23,31);(6,7)(8,9)(10,12,14,13)(11,15,16,17)(18,20)(19,25)(21,22)(23,28)(24,29)(26,30)(27,33)(31,32);(7,9)(10,13,14,12)(11,15,16,17)(18,21)(19,25)(20,22)(23,29)(24,28)(26,32)(27,33)(30,31);(11,16)(15,17)(18,22,24,23)(19,26,27,31)(20,28,29,21)(25,32,33,30)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17060);