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) = C27 |
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 | |
977 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 977 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) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(6,8)(9,10,13,11)(12,15,16,14)(17,18,20,22)(19,23,24,21);(6,8)(10,11)(12,16);(5,6,7,8)(9,11,13,10)(12,15,16,14)(17,19,20,24)(18,21,22,23);(1,2)(3,4)(9,12)(10,14)(11,15)(13,16)(17,18)(19,21)(20,22)(23,24)Polycyclic presentation:
2^8|a^2=fg,c^2=ef,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 | #125835, #125951, #126244, … |
| maximal normal subgroup of | #125835, #125951, #126244, … |
| maximal quotient of | #154686, #154853, #154867, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8494);