Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C24 |
| direct product | C5×#8966 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #120 |
| commutator | C23 | C22×C5×C8 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #386 |
| prime radical | C22×C5×D4 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
626 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)(6,7,9,12,10,8,11,13)(14,15)(16,19)(17,24,25,20)(18,21)(22,26,28,29)(23,27);(6,8,9,13,10,7,11,12)(14,16,18,23)(15,19,21,27)(17,22,25,28)(20,26,24,29);(14,17)(15,20)(16,22)(18,25)(19,26)(21,24)(23,28)(27,29)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1954);