Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1239 conjugacy classes of subgroups; only the 187 normal subgroups are shown.
This group has 1239 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) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(5,6,7,8)(15,19)(18,20);(1,2,4,3)(9,10,11,12)(14,17)(18,20);(2,3)(5,6)(7,8)(9,10)(11,12)(13,14)(15,18)(16,17)(19,20);(1,3,4,2)(13,15)(14,18)(16,19)(17,20)Polycyclic presentation:
2^8|a^2=f,b^2=egh,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de52 conjugacy classes — view the full character table.
| maximal subgroup of | #103397, #103413, #103429, … |
| maximal normal subgroup of | #103397, #103413, #103429, … |
| maximal quotient of | #155711, #155845, #155848, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 9798);