Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9239 |
| automorphism | Aut(G) = C23×D43 (order 4096) · Inn(G) = #386 · Out(G) = C26 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | C4×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
225 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 225 conjugacy classes of subgroups — too many to draw. Only its 73 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) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,5,8,6,3,4,7)(9,10,13,18,14,17,12,16)(11,15,19,22,20,23,21,24);(1,3,6,2)(4,7,5,8)(9,11,12,19,14,20,13,21)(10,15,18,24,17,23,16,22);(1,2,6,3)(4,8,5,7)(9,12,14,13)(10,16,17,18)(11,19,20,21)(15,22,23,24)Polycyclic presentation:
2^8|a^2=f,b^2=dfh,c^2=h,d^2=gh,e^2=gh,f^2=h,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=egh,f^b=fh,d^c=dgh,e^c=egh,f^c=fgh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9252, #198263, #198264, … |
| maximal normal subgroup of | C2×#9252, #198263, #198264, … |
| maximal quotient of | #99893, #99894, #99909, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2240);