Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
799 conjugacy classes of subgroups; only the 171 normal subgroups are shown.
This group has 799 conjugacy classes of subgroups — too many to draw. Only its 171 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:
(2,3)(5,6,7,8)(10,13)(11,15)(17,18,20,21)(19,22,24,23);(1,2,4,3)(10,13)(14,16)(18,21)(22,23);(2,3)(5,6)(7,8)(9,10,12,13)(11,14,15,16)(17,18)(19,23)(20,21)(22,24);(1,3,4,2)(9,11)(10,14)(12,15)(13,16)(17,19,20,24)(18,22,21,23)Polycyclic presentation:
2^8|a^2=efg,b^2=egh,c^2=eg,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de52 conjugacy classes — view the full character table.
| maximal subgroup of | #103398, #103414, #103430, … |
| maximal normal subgroup of | #103398, #103414, #103430, … |
| maximal quotient of | #155757, #155845, #155869, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 9999);