Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
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 | C25 | D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1135 conjugacy classes of subgroups; only the 567 normal subgroups are shown.
This group has 1135 conjugacy classes of subgroups — too many to draw. Only its 567 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 — 4 generators on 24 points:
(1,2)(3,4)(5,6,7,8)(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23);(2,4)(9,11,13,22,14,20,12,21)(10,15,16,24,18,23,17,19);(11,20)(15,23)(19,24)(21,22);(9,12,14,13)(10,16,18,17)(11,21,20,22)(15,24,23,19)Polycyclic presentation:
2^8|a^2=f,b^2=g,d^2=h,e^2=h,g^2=h,b^a=be,e^a=eh,g^a=gh,c^b=ch88 conjugacy classes — view the full character table.
| maximal subgroup of | #123457, #123459, #123467, … |
| maximal normal subgroup of | #123457, #123459, #123467, … |
| maximal quotient of | #180911, #180917, #180924, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13335);