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 | C2×#3147 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| socle | C23 | C22×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1683 conjugacy classes of subgroups; only the 583 normal subgroups are shown.
This group has 1683 conjugacy classes of subgroups — too many to draw. Only its 583 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 22 points:
(1,2);(4,6)(7,8)(9,16)(10,14)(11,20)(12,15)(13,18)(17,21)(19,22);(7,9,11,18,12,19,10,17)(8,13,14,16,15,21,20,22);(3,4,5,6);(7,10,12,11)(8,14,15,20)(9,17,19,18)(13,16,21,22)88 conjugacy classes — view the full character table.
| maximal subgroup of | #111342, #111377, #111536, … |
| maximal normal subgroup of | #111342, #111377, #111536, … |
| maximal quotient of | C2×#19596, C2×#19600, C2×#19603, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53620);