Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#1964 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C24 | C24 |
| prime radical | C22×#121 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
1109 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 14 points:
(1,2);(4,6)(7,8)(9,11,13,14)(10,12);(7,9)(8,11,12,14)(10,13);(3,4,5,6)64 conjugacy classes — view the full character table.
| maximal subgroup of | #105765, #107158, #107214, … |
| maximal normal subgroup of | #105765, #107158, #107214, … |
| maximal quotient of | C2×#9649, C2×#9650, C2×#9651, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14199);