Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #15257 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
649 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 649 conjugacy classes of subgroups — too many to draw. Only its 183 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,4)(10,13)(11,15)(18,21)(19,23);(6,8)(17,18,20,21)(19,24,23,22);(5,6)(7,8)(9,10,12,13)(11,14,15,16)(17,18)(19,22)(20,21)(23,24);(1,2)(3,4)(9,11)(10,14)(12,15)(13,16)(17,19,20,23)(18,22,21,24)Polycyclic presentation:
2^8|a^2=fh,b^2=e,c^2=ef,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de58 conjugacy classes — view the full character table.
| maximal subgroup of | #125826, #126016, #126151, … |
| maximal normal subgroup of | #125826, #126016, #126151, … |
| maximal quotient of | #154691, #154726, #154968, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8953);