Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#3162 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| socle | C23 | C22×D4 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
951 conjugacy classes of subgroups; only the 567 normal subgroups are shown.
This group has 951 conjugacy classes of subgroups — too many to draw. Only its 567 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(3,4,6,8)(5,9,10,7)(11,12)(13,20)(14,18)(15,24)(16,19)(17,23)(21,26)(22,25);(3,5,6,10)(4,7,8,9)(13,21)(17,25)(20,26)(22,23);(11,13,15,23,16,21,14,22)(12,17,18,20,19,25,24,26);(11,14,16,15)(12,18,19,24)(13,22,21,23)(17,20,25,26)64 conjugacy classes — view the full character table.
| maximal subgroup of | #111354, #111357, #111370, … |
| maximal normal subgroup of | #111354, #111357, #111370, … |
| maximal quotient of | C2×#20799, C2×#21154, C2×#21321, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53645);