Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23×C4 |
| direct product | C8×#99 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22⋊C4 |
| commutator | C22 | C2×C4×C8 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | C4×C22⋊C4 |
| prime radical | C22×D4 | C2×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
307 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 307 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,3,5,4,6,7,8);(9,10)(11,15,16,13)(12,14);(9,11)(10,13)(12,16)(14,15)88 conjugacy classes — view the full character table.
| maximal subgroup of | #92996, #92997, #93008, … |
| maximal normal subgroup of | #92996, #92997, #93008, … |
| maximal quotient of | C8×#323, C8×#324, #96674, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1147);