Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
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 | C25 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
477 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 477 conjugacy classes of subgroups — too many to draw. Only its 131 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10,12,17,14,16,13,18)(11,15,19,23,21,22,20,24);(1,3)(2,5)(4,8)(6,7)(9,11,13,20,14,21,12,19)(10,15,18,24,16,22,17,23);(10,16)(15,22)(17,18)(23,24)Polycyclic presentation:
2^8|a^2=e,b^2=f,e^2=h,f^2=h,b^a=bd,c^a=ch,d^a=dg,e^b=eg64 conjugacy classes — view the full character table.
| maximal subgroup of | #105996, #105997, #106079, … |
| maximal normal subgroup of | #105996, #105997, #106079, … |
| maximal quotient of | #94918, #94920, #94930, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 915);