Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17411 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | D4×#101 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
689 conjugacy classes of subgroups; only the 209 normal subgroups are shown.
This group has 689 conjugacy classes of subgroups — too many to draw. Only its 209 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4);(2,4);(5,6,8,12,9,11,17,19)(7,13,14,20,15,10,16,18);(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)55 conjugacy classes — view the full character table.
| maximal subgroup of | #138889, #138891, #138956, … |
| maximal normal subgroup of | #138889, #138891, #138956, … |
| maximal quotient of | D4×#324, #219725, #219735, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15397);