Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×C22⋊C4 |
| commutator | C23 | C22×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×C22⋊C8 |
| prime radical | C24 | C2×C8 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
243 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 243 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,6,7,11,8,12,14,18,9,10,13,16,15,17,19,20);(1,2,3,4);(6,10)(7,13)(12,17)(14,19)Polycyclic presentation:
2^8|a^2=f,b^2=d,f^2=g,g^2=h,b^a=bd,c^a=ce,e^a=eh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #93337, #93338, #93504, … |
| maximal normal subgroup of | #93337, #93338, #93504, … |
| maximal quotient of | #134218, #134246, #134273, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5773);