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 | C23×C4 | C23 |
| socle | C22 | C2×#95 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
297 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 297 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)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,20,24,22,26,21,25)(19,23,28,31,27,30,29,32);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19,21,29,22,27,20,28)(18,23,25,32,26,30,24,31);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(19,27)(23,30)(28,29)(31,32)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=g,e^2=h,f^2=h,b^a=bd,d^a=dg,c^b=ch,e^b=eg64 conjugacy classes — view the full character table.
| maximal subgroup of | #105870, #106190, #106192, … |
| maximal normal subgroup of | #105870, #106190, #106192, … |
| maximal quotient of | #94923, #94925, #94939, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 914);