Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C22×C42 |
| direct product | C4×#344 |
| subgroup | G/H | |
|---|---|---|
| center | C42 | C22⋊C4 |
| commutator | C22 | C43 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | C2×#95 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
453 conjugacy classes of subgroups; only the 227 normal subgroups are shown.
This group has 453 conjugacy classes of subgroups — too many to draw. Only its 227 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 — 3 generators on 20 points:
(1,2,3,4);(5,6)(7,13,14,10)(8,12)(9,11)(15,19,20,17)(16,18);(5,7,8,15,9,14,16,20)(6,10,12,17,11,13,18,19)88 conjugacy classes — view the full character table.
| maximal subgroup of | C8×#344, #105387, #105388, … |
| maximal normal subgroup of | C8×#344, #105387, #105388, … |
| maximal quotient of | C4×#1281, #105298, #105299, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3698);