Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10382 |
| automorphism | Aut(G) = C23×#35346 (order 2048) · Inn(G) = #393 · Out(G) = C25 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #393 |
| commutator, Frattini, cosocle | C23×C4 | C23 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
223 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 223 conjugacy classes of subgroups — too many to draw. Only its 45 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)(3,5)(4,7)(6,8)(9,10)(11,17,19,23)(12,15)(13,16)(14,20,18,21)(22,24);(3,8)(5,6)(9,10)(11,18,19,14)(12,15)(13,16)(17,20,23,21)(22,24);(2,5)(3,8)(6,7)(9,11)(10,14)(12,21)(13,19)(15,23)(16,18)(17,24)(20,22)Polycyclic presentation:
2^8|a^2=d,b^2=d,e^2=gh,b^a=bd,c^a=ce,e^a=eh,f^a=fgh,c^b=cf,e^b=eh,f^b=fg,d^c=dg,e^c=egh31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10382, #238371, #238375, … |
| maximal normal subgroup of | C2×#10382, #238371, #238375, … |
| maximal quotient of | #104753, #104756, #104761, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3370);