Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29602 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#29633 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C5×#117 ◁ G | |
1260 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)(12,15)(13,14)(16,17)(18,19)(20,29)(21,30)(22,26)(23,28)(24,27)(25,32)(31,33);(11,14)(15,17)(18,20,22,30)(19,21,26,29)(23,31,24,25)(27,33,28,32);(7,9)(10,12)(11,15)(13,16)(14,17)(18,21,22,29)(19,25,26,31)(20,28,30,27)(23,33,24,32);(12,16)(15,17)(19,26)(21,29)(27,28)(32,33)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 22621);