Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #17082 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| prime radical | C2×#392 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
937 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(9,10,12,14)(11,15,16,13)(17,18)(19,23)(20,22)(21,24);(6,8)(11,16)(13,15)(17,18,20,22)(19,23,24,21);(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,22)(21,24);(1,2)(3,4)(9,11)(10,13)(12,16)(14,15)(17,19,20,24)(18,21,22,23)Polycyclic presentation:
2^8|a^2=f,b^2=e,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,d^c=de46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#17095, #844862, #844910, … |
| maximal normal subgroup of | C2×#17095, #844862, #844910, … |
| maximal quotient of | #156467, #156472, #156474, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10083);