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×#26951 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C4×C5×C22⋊C4 ◁ G | |
884 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)(7,8)(10,11)(12,16)(13,15)(14,17)(18,19,23,27)(20,24,29,32)(21,25,22,26)(28,31,30,33);(6,7,9,8)(10,12,13,17)(11,14,15,16)(18,20)(19,24)(21,28)(22,30)(23,29)(25,31)(26,33)(27,32);(11,15)(14,16)(18,21)(19,25)(22,23)(26,27)(28,30)(31,33);(6,8,9,7)(11,15)(14,16)(18,21,23,22)(19,25,27,26)(20,28,29,30)(24,31,32,33)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 19939);