Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C23 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
415 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 415 conjugacy classes of subgroups — too many to draw. Only its 183 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)(6,8)(9,10,13,17,12,16,14,21)(11,15,19,23,18,22,20,24);(2,4)(9,11,14,19,12,18,13,20)(10,15,17,24,16,22,21,23);(5,6,7,8)(10,16)(15,22)(17,21)(23,24)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=eh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gh,d^b=dh,f^b=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #107491, #107656, #107657, … |
| maximal normal subgroup of | #107491, #107656, #107657, … |
| maximal quotient of | #108929, #108933, #108938, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3846);