Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
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 | |
789 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)(9,10,13,11)(12,15,16,14)(17,18,22,21)(19,24,20,23);(6,8)(10,11)(14,15)(17,19,22,20)(18,23,21,24);(5,6)(7,8)(9,11,13,10)(12,15,16,14)(17,20,22,19)(18,24,21,23);(1,2)(3,4)(9,12,13,16)(10,14,11,15)(17,21,22,18)(19,24,20,23)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=e,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#14744, #607679, #607756, … |
| maximal normal subgroup of | C2×#14744, #607679, #607756, … |
| maximal quotient of | #154104, #154198, #154203, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7732);