Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8746 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #381 |
| commutator, prime radical | C24 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
219 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 219 conjugacy classes of subgroups — too many to draw. Only its 59 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)(11,17,19,14)(12,20)(13,16)(15,21)(18,22,24,23);(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23);(2,5)(3,7)(9,12)(10,15,16,20)(11,18)(13,21)(14,22,17,23)(19,24)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=d,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^b=fg,e^c=eg,f^c=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #132027, #132028, #132029, … |
| maximal normal subgroup of | #132027, #132028, #132029, … |
| maximal quotient of | #99146, #99147, #99153, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1745);