Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #15257 |
| automorphism | Aut(G) order 393216 |
| 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 | C26 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1469 conjugacy classes of subgroups; only the 187 normal subgroups are shown.
This group has 1469 conjugacy classes of subgroups — too many to draw. Only its 187 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) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,4)(5,6,7,8)(11,15)(14,16)(18,21)(22,23);(6,8)(10,13)(14,16)(17,18,20,21)(19,23,24,22);(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,21)(23,24);(1,2)(3,4)(5,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,22)(20,24)(21,23)Polycyclic presentation:
2^8|a^2=fgh,b^2=f,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de58 conjugacy classes — view the full character table.
| Fitting subgroup of | #15986⋊C3 |
| maximal subgroup of | #139909, #347046, C2×#15986, … |
| maximal normal subgroup of | #139909, #347046, C2×#15986, … |
| maximal quotient of | #154676, #154802, #154922, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8974);