Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22⋊C4 |
| commutator | C22 | C22×C16 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×C22⋊C8 |
| prime radical | C22×D4 | C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
215 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 215 conjugacy classes of subgroups — too many to draw. Only its 83 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) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,6,4,5,7,8)(9,10,12,16,13,17,21,24,14,18,11,15,19,22,20,23);(2,5)(3,7);(9,11,13,20,14,12,19,21)(10,15,17,23,18,16,22,24)Polycyclic presentation:
2^8|a^2=e,c^2=fg,e^2=g,g^2=h,b^a=bd,d^a=df,e^b=ef88 conjugacy classes — view the full character table.
| maximal subgroup of | #92996, #93008, #122358, … |
| maximal normal subgroup of | #92996, #93008, #122358, … |
| maximal quotient of | #124201, #124210, #124237, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4475);