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×#1386 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #102 |
| commutator | C8 | C2×C42 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C24 | D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
315 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)(7,8,11,15)(9,10,17,19)(12,14,20,22)(13,21,18,16);(7,9,12,16,13,14,19,15,11,17,20,21,18,22,10,8)64 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#1386, #146344, #146438, … |
| maximal normal subgroup of | C4×#1386, #146344, #146438, … |
| maximal quotient of | C2×#7158, #126695, #126714, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5554);