Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) order 3145728 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C22×#428 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×D4 |
| commutator | C4 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| prime radical | C25 | D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1591 conjugacy classes of subgroups; only the 1039 normal subgroups are shown.
This group has 1591 conjugacy classes of subgroups — too many to draw. Only its 1039 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2);(3,4);(5,6,7,8)(10,13)(11,15)(14,16);(9,10,11,13,12,14,15,16);(10,14)(13,16)88 conjugacy classes — view the full character table.
| Fitting subgroup of | A4×#428 |
| maximal subgroup of | #123922, #123923, #123971, … |
| maximal normal subgroup of | #123922, #123923, #123971, … |
| maximal quotient of | C22×#1565, C22×#1569, C2×#17796, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53092);