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) = C27×C4 |
| direct product | C22×#433 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C23 |
| commutator | C22 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| socle | C24 | C22×C4 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1155 conjugacy classes of subgroups; only the 783 normal subgroups are shown.
This group has 1155 conjugacy classes of subgroups — too many to draw. Only its 783 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2);(3,4);(5,6)(7,10)(8,9)(11,12)(13,14,16,19,17,20,23,26)(15,18,22,25,21,24,27,28);(6,9)(10,12)(13,15)(14,18)(16,22)(17,21)(19,25)(20,24)(23,27)(26,28);(5,7,8,11)(6,10,9,12)(15,21)(18,24)(22,27)(25,28)112 conjugacy classes — view the full character table.
| Fitting subgroup of | A4×#433 |
| maximal subgroup of | #105563, #105573, #106147, … |
| maximal normal subgroup of | #105563, #105573, #106147, … |
| maximal quotient of | C22×#1746, C22×#1750, C22×#1768, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53104);