Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C23 | C2×C4⋊C4 |
| prime radical | C24 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
339 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 339 conjugacy classes of subgroups — too many to draw. Only its 143 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)(9,10,14,18,15,20,13,19)(11,17,21,24,22,23,12,16);(5,6,7,8)(9,11,13,12,15,22,14,21)(10,16,18,17,20,24,19,23);(2,4)(9,12,13,22,15,21,14,11)(10,17,18,24,20,23,19,16)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=d,d^2=h,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,c^b=ch64 conjugacy classes — view the full character table.
| maximal subgroup of | #93551, #93568, #128811, … |
| maximal normal subgroup of | #93551, #93568, #128811, … |
| maximal quotient of | #108514, #108515, #108518, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3811);