Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19072 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #3548 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
500 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 500 conjugacy classes of subgroups — too many to draw. Only its 115 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 — 4 generators on 24 points:
(2,5)(3,7)(6,8)(9,10,15,19)(11,20,16,23)(12,21,24,22)(13,18,14,17);(3,7)(5,8)(9,11,15,16)(10,12,19,24)(13,21,14,22)(17,23,18,20);(1,2)(3,8)(4,6)(5,7)(9,12)(10,16)(11,19)(13,20)(14,23)(15,24)(17,21)(18,22);(9,13,15,14)(10,17,19,18)(11,21,16,22)(12,23,24,20)Polycyclic presentation:
2^8|a^2=g,b^2=g,d^2=g,e^2=g,f^2=gh,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fgh,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fgh40 conjugacy classes — view the full character table.
| maximal subgroup of | #249721, #249725, #249732, … |
| maximal normal subgroup of | #249721, #249725, #249732, … |
| maximal quotient of | #168118, #168125, #168135, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12064);