Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18562 |
| automorphism | Aut(G) order 8192 |
| 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 | #3522 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
493 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 493 conjugacy classes of subgroups — too many to draw. Only its 117 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)(9,10,14,17)(11,18,20,23)(12,16,13,21)(15,22,24,19);(3,7)(5,8)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,24)(18,21);(1,2)(3,8)(4,6)(5,7)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18);(11,20)(15,24)(18,23)(19,22)Polycyclic presentation:
2^8|a^2=g,c^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #273192, #273194, #273216, … |
| maximal normal subgroup of | #273192, #273194, #273216, … |
| maximal quotient of | #166292, #166356, #166420, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11553);