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 | C22×D4 | C8 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
209 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 209 conjugacy classes of subgroups — too many to draw. Only its 71 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) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,6,4,5,7,8)(9,10,12,15,13,16,21,22,14,17,11,18,19,23,20,24);(2,5)(3,7);(9,11,13,20,14,12,19,21)(10,15,16,22,17,18,23,24)Polycyclic presentation:
2^8|a^2=e,c^2=fg,e^2=g,g^2=h,b^a=bd,c^a=ch,d^a=df,e^b=ef64 conjugacy classes — view the full character table.
| maximal subgroup of | #93000, #93004, #122362, … |
| maximal normal subgroup of | #93000, #93004, #122362, … |
| maximal quotient of | #124202, #124211, #124238, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4476);