Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#1859 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
783 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);(3,4,6,5)(8,12)(9,10)(13,14);(3,5,6,4)(7,8)(9,14)(10,12)(11,13);(7,8)(9,12)(10,14)(11,13)64 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#1859, #354447, #356724, … |
| maximal normal subgroup of | C4×#1859, #354447, #356724, … |
| maximal quotient of | C2×#8839, C2×#8841, #191816, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13849);