Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
| direct product | C22×#495 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | C23×D8 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1759 conjugacy classes of subgroups; only the 607 normal subgroups are shown.
This group has 1759 conjugacy classes of subgroups — too many to draw. Only its 607 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2);(3,4);(6,8)(9,10)(11,17)(12,20)(13,15)(14,18)(16,21)(19,23)(22,24);(9,11,12,18,13,19,21,24)(10,14,16,17,15,22,20,23);(5,6,7,8)(10,15)(14,22)(16,20)(17,23)88 conjugacy classes — view the full character table.
| Fitting subgroup of | A4×#495 |
| maximal subgroup of | #111412, #129989, #147864, … |
| maximal normal subgroup of | #111412, #129989, #147864, … |
| maximal quotient of | C22×#1706, C22×#1707, C22×#1708, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53619);