Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25 |
| direct product | C2×#2110 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C42 |
| commutator | C4 | C22×C42 |
| Frattini, cosocle | C2×C8 | C24 |
| socle | C22 | C2×#97 |
| prime radical | C24 | M16 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
369 conjugacy classes of subgroups; only the 303 normal subgroups are shown.
This group has 369 conjugacy classes of subgroups — too many to draw. Only its 303 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,6,10,7,12,18,26,8,11,19,25,20,27,33,34)(5,13,15,29,16,22,31,24,14,28,17,32,30,9,21,23);(3,5)(4,9,11,22)(6,17)(7,16)(8,14)(10,24,25,23)(12,13,27,28)(15,19)(18,21)(20,30)(26,32,34,29)(31,33);(5,14)(9,22)(13,28)(15,17)(16,30)(21,31)(23,24)(29,32)88 conjugacy classes — view the full character table.
| maximal subgroup of | #127691, #127692, #127693, … |
| maximal normal subgroup of | #127691, #127692, #127693, … |
| maximal quotient of | C2×#11405, C2×#11406, #479910, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26316);