Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#1735 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | D4 |
| commutator | C4 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C24 | C2×D4 |
| prime radical | C25 | D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1199 conjugacy classes of subgroups; only the 607 normal subgroups are shown.
This group has 1199 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) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4)(5,6)(7,8,9,10)(11,12)(13,21)(14,18)(15,19)(16,20)(17,22)(23,26)(24,25);(4,6)(11,13,15,23,16,24,14,22)(12,17,18,25,20,26,19,21);(11,14,16,15)(12,18,20,19)(13,22,24,23)(17,25,26,21)112 conjugacy classes — view the full character table.
| maximal subgroup of | #123436, #123437, #123443, … |
| maximal normal subgroup of | #123436, #123437, #123443, … |
| maximal quotient of | C2×#7872, C2×#7875, #180915, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13333);