Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C4×#509 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D8 |
| commutator | C8 | C23×C4 |
| Frattini, cosocle | C2×C8 | C24 |
| socle | C22 | C22×D8 |
| prime radical | C2×#509 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
369 conjugacy classes of subgroups; only the 161 normal subgroups are shown.
This group has 369 conjugacy classes of subgroups — too many to draw. Only its 161 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);(5,6)(7,10)(8,17)(9,11)(12,14)(13,15)(16,18)(19,20);(6,10)(7,14)(8,15)(11,18)(12,13)(16,20)(17,19);(6,11)(10,18)(12,19)(13,17)64 conjugacy classes — view the full character table.
| maximal subgroup of | #136693, #136695, #136698, … |
| maximal normal subgroup of | #136693, #136695, #136698, … |
| maximal quotient of | C8×#509, #483784, #483786, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26440);