Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 1572864 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
| direct product | C22×#446 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C23 |
| commutator | C22 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| socle, prime radical | C24 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1111 conjugacy classes of subgroups; only the 863 normal subgroups are shown.
This group has 1111 conjugacy classes of subgroups — too many to draw. Only its 863 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2);(3,4);(5,6)(7,12)(8,10)(9,11)(13,14,16,15,17,18,20,19);(5,7,9,8)(6,10,11,12);(5,8,9,7)(6,10,11,12)(13,15,16,18,17,19,20,14)112 conjugacy classes — view the full character table.
| Fitting subgroup of | A4×#446 |
| maximal subgroup of | #105579, #105590, #105593, … |
| maximal normal subgroup of | #105579, #105590, #105593, … |
| maximal quotient of | C22×#1771, C22×#1770, C2×C4×#446, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53174);