Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #2263 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#2263 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D16 |
| commutator, Frattini, cosocle | C16 | C24 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
251 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 251 conjugacy classes of subgroups — too many to draw. Only its 103 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)(5,9)(6,18)(7,21)(8,10)(11,14)(12,15)(13,19)(16,28)(17,22)(20,34)(23,30)(24,29)(25,31)(26,32)(27,33);(4,9)(5,14)(6,15)(7,19)(10,22)(11,12)(13,28)(16,33)(17,31)(18,23)(20,32)(21,24)(25,26)(27,34)(29,30);(4,10)(9,22)(11,25)(12,26)(13,21)(18,34)(23,27)(24,28)52 conjugacy classes — view the full character table.
| maximal subgroup of | #152531, #152534, #152538, … |
| maximal normal subgroup of | #152531, #152534, #152538, … |
| maximal quotient of | C2×#13617, C2×#13616, C2×#13634, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26967);