Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C29 |
| direct product | C22×#504 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C16 | C22 |
| commutator | C2 | C24×C8 |
| Frattini, cosocle | C8 | C25 |
| socle | C23 | C22×C8 |
| prime radical | C22×#41 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
1059 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2);(3,4);(5,6,8,12,9,13,18,24,10,14,19,25,20,26,30,34)(7,11,16,22,17,23,29,33,15,21,27,31,28,32,35,36);(5,7)(6,11)(8,16)(9,17)(10,15)(12,22)(13,23)(14,21)(18,29)(19,27)(20,28)(24,33)(25,31)(26,32)(30,35)(34,36);(7,15)(11,21)(16,27)(17,28)(22,31)(23,32)(29,35)(33,36)160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55610);