Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11663 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#102 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C24 | SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
221 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 221 conjugacy classes of subgroups — too many to draw. Only its 75 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)(9,10,12,16,13,18,23,17)(11,19,21,15,22,14,20,24);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(10,14)(11,20)(12,23)(15,16)(17,24)(18,19)(21,22)Polycyclic presentation:
2^8|a^2=f,c^2=d,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gh,e^c=egh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #130802, #130803, #130804, … |
| maximal normal subgroup of | #130802, #130803, #130804, … |
| maximal quotient of | #135254, #135255, #135284, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5859);