Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) order 786432 |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C25×C4 |
| direct product | C22×#322 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C42 | C22 |
| commutator | C2 | C22×C4×C8 |
| Frattini, cosocle | C42 | C24 |
| socle, prime radical | C24 | C42 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
591 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2);(3,4);(5,6,7,10,8,9,11,12);(6,9)(10,12)(13,14,15,17,16,18,19,20)160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10299);