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 | C2×#3529 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator, Frattini, cosocle | C22 | C26 |
| socle | C23 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
3709 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 26 points:
(1,2);(3,4)(5,7)(6,8)(9,10)(11,12)(13,19)(14,18)(15,17)(16,20)(21,25)(22,24)(23,26);(3,5)(4,7)(6,9)(8,10)(11,13)(12,16)(14,21)(15,20)(17,19)(18,23)(22,26)(24,25);(4,8)(5,9)(13,20)(16,19)(21,26)(23,25);(4,8)(7,10)(12,17)(13,20)(18,24)(21,26);(11,14,15,22)(12,18,17,24)(13,21,20,26)(16,23,19,25)76 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55679);