Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C24 |
| commutator | C23 | C23×C4 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
959 conjugacy classes of subgroups.
| 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)(19,21)(20,22)(23,24);(10,13)(14,16)(18,19)(20,24);(1,2,4,3)(9,10)(11,14)(12,13)(15,16)(17,19,21,18)(20,23,24,22);(1,3,4,2)(9,11)(10,14)(12,15)(13,16)(17,20,21,24)(18,22,19,23)Polycyclic presentation:
2^8|a^2=h,c^2=efg,d^2=efg,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de64 conjugacy classes — view the full character table.
| maximal subgroup of | #148167, #148816, #149030, … |
| maximal normal subgroup of | #148167, #148816, #149030, … |
| maximal quotient of | #153962, #153976, #154012, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7501);