Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
| direct product | C4×#416 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | D4 |
| commutator | C4 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C22×D4 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
835 conjugacy classes of subgroups; only the 367 normal subgroups are shown.
This group has 835 conjugacy classes of subgroups — too many to draw. Only its 367 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 — 4 generators on 24 points:
(1,2,3,4);(5,6,7,8)(9,10)(11,18)(12,16)(13,21)(14,17)(15,19)(20,23)(22,24);(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,24)(18,21);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18)112 conjugacy classes — view the full character table.
| maximal subgroup of | #106412, #106413, #107048, … |
| maximal normal subgroup of | #106412, #106413, #107048, … |
| maximal quotient of | C4×#1476, #182154, #182169, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13396);