Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4×C5 · Schur multiplier (H2) = C214 |
| direct product | C2×C5×#3428 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C24 |
| commutator | C2 | C25×C4×C5 |
| Frattini, cosocle | C22 | C26×C5 |
| socle | C23×C5 | C25 |
| prime radical | C23×C5×#41 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ G | |
11534 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 6 generators on 27 points:
(1,2)(23,24,25,26,27);(3,4,5,6)(7,8)(9,15)(10,14)(11,13)(12,16)(17,21)(18,20)(19,22);(7,9)(8,12)(10,17)(11,16)(13,15)(14,19)(18,22)(20,21);(9,16)(12,15)(17,22)(19,21);(8,13)(12,15)(14,20)(19,21);(7,10,11,18)(8,14,13,20)(9,17,16,22)(12,19,15,21)680 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 55636);