Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12948 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#12949 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1864 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
394 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,11,18,13,19,12,30)(8,20,22,34,24,31,23,35)(9,25,28,17,10,14,26,16)(15,27,32,21,33,29,36,37);(7,14,19,25)(9,26)(10,28)(11,12)(15,31,33,20)(16,18,17,30)(21,27)(22,23)(29,37)(32,34,36,35);(6,8)(7,15)(9,27)(10,29)(11,23)(12,22)(13,24)(14,31)(16,34)(17,35)(18,36)(19,33)(20,25)(21,26)(28,37)(30,32)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5937);