Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D8 |
| commutator | C8 | C2×C42 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
263 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 263 conjugacy classes of subgroups — too many to draw. Only its 103 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)(5,6,8,7)(10,15)(11,19)(12,21)(13,20)(16,22)(17,18)(23,24);(2,4)(9,10,12,17,13,18,11,15,14,16,19,24,20,23,21,22);(5,7,8,6)(10,16)(15,22)(17,24)(18,23)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=e,d^2=gh,f^2=g,g^2=h,b^a=bd,d^a=dg,f^a=fgh,g^a=gh,c^b=ch64 conjugacy classes — view the full character table.
| maximal subgroup of | #128688, #128719, #128738, … |
| maximal normal subgroup of | #128688, #128719, #128738, … |
| maximal quotient of | #128531, #128585, #128600, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5625);