Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | Aut(G) order 262144 |
| 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 | |
889 conjugacy classes of subgroups; only the 191 normal subgroups are shown.
This group has 889 conjugacy classes of subgroups — too many to draw. Only its 191 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)(6,7)(17,18)(19,23)(20,21)(22,24);(5,6,8,7)(10,13)(14,16)(18,21)(22,23);(1,2,3,4)(5,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,21)(23,24);(9,11,12,15)(10,14,13,16)(17,19)(18,22)(20,24)(21,23)Polycyclic presentation:
2^8|b^2=eh,c^2=f,d^2=gh,b^a=be,c^a=cf,d^a=dh,c^b=cg64 conjugacy classes — view the full character table.
| maximal subgroup of | #102969, #102979, #102989, … |
| maximal normal subgroup of | #102969, #102979, #102989, … |
| maximal quotient of | #154192, #154413, #154431, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8238);