Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C4 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C22×D4 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
571 conjugacy classes of subgroups; only the 351 normal subgroups are shown.
This group has 571 conjugacy classes of subgroups — too many to draw. Only its 351 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 — 4 generators on 24 points:
(1,2,4,3)(9,10,13,18,14,16,12,17)(11,19,20,15,22,24,21,23);(5,6,7,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,23)(17,24)(18,19);(1,3,4,2)(10,16)(15,23)(17,18)(19,24);(9,12,14,13)(10,17,16,18)(11,20,22,21)(15,24,23,19)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=ef,d^2=h,e^2=h,f^2=h,b^a=be,c^a=ch,e^b=eh,f^b=fh88 conjugacy classes — view the full character table.
| maximal subgroup of | #116957, #116963, #117707, … |
| maximal normal subgroup of | #116957, #116963, #117707, … |
| maximal quotient of | #183364, #183371, #183379, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13439);