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×C42 |
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 | |
865 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 865 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) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,8)(14,15)(16,20);(6,8)(10,12)(13,14,17,15)(16,19,20,18);(9,10,11,12)(13,15,17,14)(16,19,20,18);(1,2)(3,4)(5,6)(7,8)(13,16,17,20)(14,18,15,19)Polycyclic presentation:
2^8|b^2=f,c^2=fg,d^2=f,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 | #125784, #126100, #126193, … |
| maximal normal subgroup of | #125784, #126100, #126193, … |
| maximal quotient of | #154782, #154785, #154882, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8768);