Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7408 |
| automorphism | Aut(G) = C23×#2196 (order 1024) · Inn(G) = AΣL(1,𝔽16) · Out(G) = C24 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | AΣL(1,𝔽16) |
| commutator | C2×C4⋊C4 | C2×C4 |
| Frattini, cosocle | #384 | C22 |
| prime radical | #537 | C4 |
| LCS | G ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #384 ◁ G | |
| LECS | G ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
114 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 114 conjugacy classes of subgroups — too many to draw. Only its 23 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 — 2 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,18,19,23,20,24,21,14)(12,15)(13,17)(16,22);(2,5)(3,8)(6,7)(9,11)(10,14)(12,21)(13,20)(15,18)(16,24)(17,23)(19,22)Polycyclic presentation:
2^8|a^2=d,c^2=f,d^2=gh,e^2=h,b^a=bc,c^a=ce,e^a=eg,c^b=cf,d^b=defh,e^b=eh,g^b=gh,d^c=dgh,e^c=eh31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7417, #150237, #150238, … |
| maximal normal subgroup of | C2×#7417, #150237, #150238, … |
| maximal quotient of | #93813, #93821, #93844, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 405);