Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) order 3145728 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C28 |
| direct product | C22×#340 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | D4 |
| commutator | C4 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1503 conjugacy classes of subgroups; only the 863 normal subgroups are shown.
This group has 1503 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 — 4 generators on 20 points:
(1,2);(3,4);(5,6)(7,8)(9,10,11,12)(14,18)(15,16)(19,20);(6,8)(13,14,16,19,17,20,15,18)112 conjugacy classes — view the full character table.
| Fitting subgroup of | A4×#340 |
| maximal subgroup of | #123432, #123433, #123447, … |
| maximal normal subgroup of | #123432, #123433, #123447, … |
| maximal quotient of | C22×#1277, C2×#7874, C2×#7877, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13332);