Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10557 |
| automorphism | Aut(G) = C23×D4×#120 (order 2048) · Inn(G) = #394 · Out(G) = C25 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #394 |
| 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 | |
237 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 237 conjugacy classes of subgroups — too many to draw. Only its 47 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) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,17)(11,18,19,15)(12,14,21,16)(20,23,24,22);(1,3)(2,5)(4,7)(6,8)(10,14)(11,19)(12,21)(15,23)(16,17)(18,22);(5,8)(9,11,12,20,13,19,21,24)(10,15,16,23,17,18,14,22)Polycyclic presentation:
2^8|a^2=g,c^2=d,d^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fh,c^b=cf,d^b=dg,e^b=eh,e^c=eg,f^c=fg34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10558, #239306, #239308, … |
| maximal normal subgroup of | C2×#10558, #239306, #239308, … |
| maximal quotient of | #104859, #104860, #104865, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3546);