Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9914 |
| automorphism | Aut(G) = C27×#120 (order 4096) · Inn(G) = #120 · Out(G) = C27 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator, Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
258 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 65 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:
(1,2,5,7)(3,8,4,6)(9,10)(11,14)(12,15)(13,16)(17,18)(19,21)(20,22)(23,24);(1,3,5,4)(2,6,7,8)(9,11,13,12)(10,14,16,15)(17,19)(18,21)(20,23)(22,24);(1,4,5,3)(2,6,7,8)(10,15)(11,12)(14,16)(21,24)Polycyclic presentation:
2^8|a^2=d,b^2=dg,c^2=d,e^2=g,b^a=bd,c^a=ce,e^a=eg,f^a=fh,c^b=cf,e^b=eh,e^c=eg40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10571, #241129, #241132, … |
| maximal normal subgroup of | C2×#10571, #241129, #241132, … |
| maximal quotient of | #105173, #105175, #105177, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3559);