Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×C4×#101 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22⋊C4 |
| commutator | C22 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C2×C22⋊C4 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
769 conjugacy classes of subgroups; only the 359 normal subgroups are shown.
This group has 769 conjugacy classes of subgroups — too many to draw. Only its 359 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(3,4,5,6);(7,8,10,14,11,13,19,21)(9,15,16,22,17,12,18,20);(7,9,11,17)(8,12,13,15)(10,18,19,16)(14,22,21,20)88 conjugacy classes — view the full character table.
| maximal subgroup of | #105577, #105579, #105613, … |
| maximal normal subgroup of | #105577, #105579, #105613, … |
| maximal quotient of | C2×C4×#324, C4×#1461, C2×#8154, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13383);