Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×C4×M32 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C8 | C22 |
| commutator | C2 | C22×C4×C8 |
| Frattini, cosocle | C2×C8 | C24 |
| socle | C23 | C22×C8 |
| prime radical | C24 | C2×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
459 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(3,4,5,6);(7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22);(8,16)(10,18)(12,20)(14,22)160 conjugacy classes — view the full character table.
| maximal subgroup of | #126547, #126548, #126550, … |
| maximal normal subgroup of | #126547, #126548, #126550, … |
| maximal quotient of | C2×C8×M32, C2×#11402, C2×#11409, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26310);