Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
| direct product | C4×#490 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C22×D4 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
485 conjugacy classes of subgroups; only the 179 normal subgroups are shown.
This group has 485 conjugacy classes of subgroups — too many to draw. Only its 179 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) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4);(5,6)(7,10)(8,9)(11,12)(14,19)(15,20)(16,17);(6,9)(10,12)(13,14,16,19,18,20,17,15);(5,7,8,11)(6,10,9,12)(13,15,16,14,18,19,17,20)64 conjugacy classes — view the full character table.
| maximal subgroup of | #99538, #99544, #124205, … |
| maximal normal subgroup of | #99538, #99544, #124205, … |
| maximal quotient of | C8×#490, #212619, #212625, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14638);