Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×#2243 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
317 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 317 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(11,12)(13,22)(14,19)(15,18)(16,24)(17,21)(20,23)(25,26);(4,7)(8,10)(11,12,15,19,16,20,22,26,17,21,25,13,23,24,14,18);(3,5,6,9)(4,8,7,10)(11,13,14,21,23,26,25,20,17,19,22,12,16,18,15,24)52 conjugacy classes — view the full character table.
| maximal subgroup of | #137510, #137511, #141549, … |
| maximal normal subgroup of | #137510, #137511, #141549, … |
| maximal quotient of | C2×#12890, C2×#13149, C2×#13150, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26686);