Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7606 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
228 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 85 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) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5)(3,7)(6,8)(9,10,12,14,13,16,11,15)(17,18,20,22,21,23,19,24);(1,2,4,6)(3,8,7,5)(10,14)(11,12)(15,16);(17,19,21,20)(18,22,23,24)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=fgh,d^2=g,e^2=h,b^a=bd,c^a=cfgh,d^a=df,d^b=dg,e^b=efg52 conjugacy classes — view the full character table.
| maximal subgroup of | #98283, #98285, #98288, … |
| maximal normal subgroup of | #98283, #98285, #98288, … |
| maximal quotient of | #96265, #96743, #96922, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 640);