Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12324 |
| automorphism | Aut(G) = C23×#21452 (order 2048) · Inn(G) = #447 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #447 |
| commutator, Frattini, cosocle | C4×C8 | C23 |
| prime radical | #1687 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #436 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
151 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 151 conjugacy classes of subgroups — too many to draw. Only its 39 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) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,3)(4,5)(7,8)(10,15)(11,14)(12,19)(16,23)(17,24)(18,21)(20,22);(1,2,6,8)(3,4,7,5)(9,10,13,17)(11,16,14,22)(12,20,21,23)(15,18,24,19);(1,3)(2,4)(5,8)(6,7)(9,11,13,14)(10,16,17,22)(12,19,21,18)(15,23,24,20)Polycyclic presentation:
2^8|b^2=g,c^2=h,d^2=fh,e^2=gh,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=egh,f^a=fh,d^b=df,e^b=eg,f^b=fh,d^c=dg,e^c=egh31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12349, #489012, #489014, … |
| maximal normal subgroup of | C2×#12349, #489012, #489014, … |
| maximal quotient of | #125678, #125680, #125714, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5337);