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) = C25×C4 |
| direct product | C4×C8×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C8 | C22 |
| commutator | C2 | C22×C4×C8 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C23×C4 |
| prime radical | C22×D4 | C2×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
711 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,3,4);(5,6,7,8,9,10,11,12);(13,14,15,16);(14,16)160 conjugacy classes — view the full character table.
| maximal subgroup of | C8×D4⋊C8, #96686, #96688, … |
| maximal normal subgroup of | C8×D4⋊C8, #96686, #96688, … |
| maximal quotient of | C82×D4, #183838, C8×#436, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13471);