Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1685 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#9238 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #121 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle | C22×C5 | #386 |
| prime radical | C5×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ G | |
512 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5)(6,7,10,13,11,8,9,12)(14,15,18,21,19,22,17,27)(16,23,26,28,24,29,25,20);(6,8,11,7)(9,12,10,13)(14,16)(15,20,22,28)(17,25)(18,26)(19,24)(21,23,27,29);(6,7,11,8)(9,13,10,12)(16,24)(20,28)(23,29)(25,26)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2226);