Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7694 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#9154 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| prime radical | C25×C5 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C24×C4×C5 ◁ G | |
902 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 33 points:
(1,2,3,4,5)(6,7)(8,9)(10,11)(12,14,16,17)(13,15)(18,19,22,26,23,27,21,25)(20,28,29,32,31,33,30,24);(10,12)(11,14,15,17)(13,16)(18,20)(19,24,27,32)(21,30)(22,29)(23,31)(25,33,26,28);(7,9)(12,16)(14,17)(18,21,23,22)(19,25,27,26)(20,29,31,30)(24,28,32,33)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2142);