Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29730 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
409 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 409 conjugacy classes of subgroups — too many to draw. Only its 143 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:
(5,6,9,10,8,12,7,11)(13,14,18,20,17,15,16,19);(1,2)(3,4)(14,15)(19,20);(2,4)(6,10)(7,9)(11,12)(13,15,17,14)(16,19,18,20);(13,16,17,18)(14,19,15,20)Polycyclic presentation:
2^8|a^2=fh,c^2=e,d^2=e,f^2=h,b^a=be,c^a=cf,c^b=cg,d^c=de,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #271010, #271014, #271026, … |
| maximal normal subgroup of | #271010, #271014, #271026, … |
| maximal quotient of | #330736, #330740, #330742, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22865);