Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#12961 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | B(4,𝔽2) |
| commutator | C2×D4 | C22×C4×C5 |
| Frattini, cosocle | C22×D4 | C23×C5 |
| prime radical | C22×C5×D4 | C23 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C5 ◁ C22×C5×D4 ◁ G | |
504 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3,4,5)(6,7,9,8)(10,11)(12,16)(13,15)(14,17)(18,25)(19,22)(20,24)(21,23);(6,8,9,7)(11,15)(12,18)(16,24)(17,23)(19,20)(22,25);(10,12,14,18)(11,16,17,25)(13,20,21,19)(15,22,23,24)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5949);