Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×C22⋊C4 |
| commutator | C23 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | C2×#95 |
| prime radical | C24 | C42 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
276 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 276 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9,10,6)(4,8)(5,7)(11,15,16,13)(12,14)(17,18,21,25,22,26,20,24)(19,23,27,30,28,31,29,32);(1,3,4,11,5,10,12,16)(2,6,8,13,7,9,14,15)(17,19)(18,23)(20,29)(21,27)(22,28)(24,32)(25,30)(26,31);(17,20,22,21)(18,24,26,25)(19,27,28,29)(23,30,31,32)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=h,e^2=h,f^2=g,b^a=bd,d^a=dg,c^b=ch,e^b=eg64 conjugacy classes — view the full character table.
| maximal subgroup of | #105759, #105870, #106083, … |
| maximal normal subgroup of | #105759, #105870, #106083, … |
| maximal quotient of | #94924, #94925, #94926, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 910);