Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
| direct product | C2×#1383 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #106 |
| 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) = 64 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2);(3,4)(5,6)(8,13,14,15)(9,17)(10,19)(11,18)(16,22,20,21);(4,6)(7,8,10,15,11,16,19,22,12,14,9,13,18,20,17,21)64 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#1383, #146344, #146426, … |
| maximal normal subgroup of | C4×#1383, #146344, #146426, … |
| maximal quotient of | C2×#7156, #126654, #126690, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5550);