Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
| direct product | C4×#357 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | D8 |
| commutator | C8 | C2×C42 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C23 | C2×D8 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
267 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 267 conjugacy classes of subgroups — too many to draw. Only its 111 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 — 3 generators on 24 points:
(1,2,3,4);(5,6,7,8)(9,10)(11,14)(12,21)(13,17)(15,18)(16,19)(20,23)(22,24);(10,14)(11,18)(12,19)(15,16)(17,23)(20,24)(21,22)88 conjugacy classes — view the full character table.
| maximal subgroup of | C8×#357, #128666, #128704, … |
| maximal normal subgroup of | C8×#357, #128666, #128704, … |
| maximal quotient of | C4×#1331, #128587, #128598, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5623);