Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
| direct product | C2×#1387 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #102 |
| commutator | C8 | C2×C42 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C24 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
307 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2);(3,4,5,6)(8,12)(9,16)(10,17)(13,15)(14,21)(18,22)(19,20);(7,8,9,13,17,19,22,20,11,15,18,12,10,14,16,21)64 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#1387, #146344, #146436, … |
| maximal normal subgroup of | C4×#1387, #146344, #146436, … |
| maximal quotient of | C2×#7159, #126697, #126716, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5556);