Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#12957 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #442 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C23×C5 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C2×C5×#97 ◁ G | |
396 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(6,7)(8,9)(10,11,15,22,17,23,16,34)(12,24,27,40,29,38,28,35)(13,21,31,39,32,18,14,20)(19,33,36,26,37,25,41,30);(7,9)(11,18,23,21)(13,14)(15,16)(19,35,37,40)(20,22,39,34)(24,36,38,41)(25,26)(27,28)(30,33)(31,32);(10,12,17,29)(11,19,23,37)(13,30,32,26)(14,33,31,25)(15,28,16,27)(18,35,21,40)(20,38,39,24)(22,41,34,36)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5945);