Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C214 |
| direct product | C4×#585 |
| subgroup | G/H | |
|---|---|---|
| center | C42 | C24 |
| commutator | C2 | C25×C4 |
| Frattini, socle, cosocle | C22 | C26 |
| prime radical | C2×#585 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
5673 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 20 points:
(1,2,3,4);(5,6)(7,13)(8,12)(9,11)(10,14)(15,19)(16,18)(17,20);(5,7)(6,10)(8,15)(9,14)(11,13)(12,17)(16,20)(18,19);(7,14)(10,13)(15,20)(17,19);(6,11)(10,13)(12,18)(17,19);(5,8,9,16)(6,12,11,18)(7,15,14,20)(10,17,13,19)136 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55639);