Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| prime radical | C25 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
723 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(6,8)(9,10,12,14)(11,15,16,13)(17,18,20,22)(19,23,24,21);(1,2,4,3)(11,16)(13,15)(17,18)(19,21)(20,22)(23,24);(2,3)(5,6,7,8)(9,11,12,16)(10,13,14,15)(17,18)(19,23)(20,22)(21,24);(1,3,4,2)(9,10)(11,13)(12,14)(15,16)(17,19,20,24)(18,21,22,23)Polycyclic presentation:
2^8|a^2=eg,b^2=egh,c^2=ef,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 | #258079, #258085, #258091, … |
| maximal normal subgroup of | #258079, #258085, #258091, … |
| maximal quotient of | #155717, #155843, #155852, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 9889);