Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C4×C16 · Schur multiplier (H2) = C23×C4 |
| direct product | C16×C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C16 | C22 |
| commutator | C2 | C2×C4×C16 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C23 | C22×C8 |
| prime radical | C24 | C2×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
255 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 255 conjugacy classes of subgroups — too many to draw. Only its 143 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:
(9,10,11,14,12,15,17,20,13,16,18,21,19,22,23,24);(1,2)(3,4)(5,6,7,8);(2,4)160 conjugacy classes — view the full character table.
| maximal subgroup of | #94122, #94125, C16×#99, … |
| maximal normal subgroup of | #94122, #94125, C16×#99, … |
| maximal quotient of | C16×C22⋊C8, #128263, #128282, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5598);