Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C22×#564 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C24 |
| commutator, Frattini, cosocle | C22 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
3187 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 36 points:
(3,4);(1,2)(3,4);(7,13)(8,16)(10,14)(11,18)(21,22,25,28)(23,26,31,29)(24,32,33,27)(30,35,36,34);(5,6,9,12)(7,14,13,10)(8,11,16,18)(15,20,19,17)(21,22)(23,29)(24,32)(25,28)(26,31)(27,33)(30,34)(35,36);(5,7,9,13)(6,10,12,14)(8,15,16,19)(11,17,18,20)(21,23)(22,26)(24,30)(25,31)(27,34)(28,29)(32,35)(33,36);(5,8)(6,11)(7,15)(9,16)(10,17)(12,18)(13,19)(14,20)(21,24,25,33)(22,27,28,32)(23,30,31,36)(26,34,29,35)76 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55680);