Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | Aut(G) order 49152 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25 |
| direct product | C22×#347 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C42 |
| commutator | C4 | C22×C42 |
| Frattini, cosocle | C2×C8 | C24 |
| socle | C23 | #97 |
| prime radical | C24 | M16 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
391 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2);(3,4);(5,6,7,11,8,13,15,19,9,12,14,20,16,10,17,18);(6,10,12,13)(7,14)(11,19,20,18)(15,17)88 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26315);