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) = 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 | |
775 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)(9,10,13,12)(11,14,15,16)(18,19)(20,24);(6,8)(10,12)(11,15)(17,18,21,19)(20,22,24,23);(5,6)(7,8)(9,11)(10,14)(12,16)(13,15)(17,19,21,18)(20,22,24,23);(1,2)(3,4)(9,12,13,10)(11,14,15,16)(17,20,21,24)(18,22,19,23)Polycyclic presentation:
2^8|a^2=ef,b^2=f,c^2=f,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×#17126, #844897, #844909, … |
| maximal normal subgroup of | C2×#17126, #844897, #844909, … |
| maximal quotient of | #156473, #156484, #156485, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10114);