Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
| direct product | C2×#1332 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | D8 |
| commutator | C8 | C22×C8 |
| Frattini, cosocle | C4×C8 | C23 |
| socle, prime radical | C23 | #102 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
203 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 203 conjugacy classes of subgroups — too many to draw. Only its 95 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) = 256 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2);(3,4,5,7,6,8,9,10)(12,16)(13,19)(14,21)(17,26)(18,25)(20,23)(22,24);(11,12,15,18)(13,20,22,26)(14,23,21,17)(16,24,25,19)88 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#1332, #128532, #128586, … |
| maximal normal subgroup of | C4×#1332, #128532, #128586, … |
| maximal quotient of | C2×#7069, #122445, #122451, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4578);