Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9519 |
| automorphism | Aut(G) = C23×D4×#120 (order 2048) · Inn(G) = #386 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | #3003 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
386 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 386 conjugacy classes of subgroups — too many to draw. Only its 61 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)(10,14)(11,18)(12,15)(17,22)(19,20)(21,23);(1,2)(3,5)(4,6)(7,8)(9,10,15,22,13,17,12,14)(11,16,18,23,20,24,19,21);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,19)(13,20)(14,21)(15,18)(17,24)(22,23)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=egh,e^b=eh,d^c=dh,e^c=eg,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9520, #201910, #201915, … |
| maximal normal subgroup of | C2×#9520, #201910, #201915, … |
| maximal quotient of | #100221, #100222, #100229, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2508);