Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) order 786432 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C22×#430 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C4 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| socle | C23 | C2×C4⋊C4 |
| prime radical | C25 | Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1359 conjugacy classes of subgroups; only the 999 normal subgroups are shown.
This group has 1359 conjugacy classes of subgroups — too many to draw. Only its 999 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) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2);(3,4);(5,6,9,12,10,13,8,18)(7,14,17,19,15,20,16,11);(5,7,8,16,10,15,9,17)(6,11,12,14,13,19,18,20);(7,15)(11,19)(14,20)(16,17)88 conjugacy classes — view the full character table.
| Fitting subgroup of | A4×#430 |
| maximal subgroup of | #123894, #123895, #123898, … |
| maximal normal subgroup of | #123894, #123895, #123898, … |
| maximal quotient of | C22×#1565, C2×#17787, C22×#1569, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53099);