Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = C25×C4×D4 (order 1024) · Inn(G) = D4 · Out(G) = C25×C4 |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C22 |
| direct product | C16×SD16 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | D4 |
| commutator | C4 | C22×C16 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C22 | C8×D4 |
| prime radical | C2×D4 | C2×C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
119 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 119 conjugacy classes of subgroups — too many to draw. Only its 67 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:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16);(17,18,19,20,21,22,23,24);(18,20)(19,23)(22,24)112 conjugacy classes — view the full character table.
| maximal subgroup of | SD16×C32, #151088, #151089, … |
| maximal normal subgroup of | SD16×C32, #151088, #151089, … |
| maximal quotient of | C16×#102, C16×#103, C16×#106, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5036);