Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25 |
| direct product | C2×#1451 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C23 |
| commutator | C22 | C22×C42 |
| Frattini, cosocle | C42 | C24 |
| socle | C23 | C2×C42 |
| prime radical | C24 | C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
483 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4,5,8,6,7,9,10)(12,15)(16,18);(11,12,13,16,14,15,17,18);(4,7)(8,10)112 conjugacy classes — view the full character table.
| maximal subgroup of | #105261, #105264, #105290, … |
| maximal normal subgroup of | #105261, #105264, #105290, … |
| maximal quotient of | C2×#7842, C4×#1451, C2×#7848, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10305);