Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | Aut(G) order 262144 |
| 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 | |
1075 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(10,13)(14,16)(17,18)(19,21)(20,22)(23,24);(6,8)(18,19)(22,23);(1,2,3,4)(5,6,7,8)(9,10,12,13)(11,14,15,16)(17,19,21,18)(20,22,24,23);(9,11,12,15)(10,14,13,16)(17,20,21,24)(18,22,19,23)Polycyclic presentation:
2^8|c^2=h,d^2=f,b^a=be,c^a=cf,c^b=cg,d^c=de64 conjugacy classes — view the full character table.
| maximal subgroup of | #106798, #106799, #107116, … |
| maximal normal subgroup of | #106798, #106799, #107116, … |
| maximal quotient of | #153936, #153937, #153941, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7332);