Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3186 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
411 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 411 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) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(6,9)(7,11)(14,17)(15,19)(18,20);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(13,14,16,17)(15,19,20,18);(6,9)(10,12)(13,15,14,18,16,20,17,19);(5,7,8,11)(6,10,9,12)(13,14,16,17)(15,18,20,19)Polycyclic presentation:
2^8|a^2=g,b^2=gh,c^2=fh,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce64 conjugacy classes — view the full character table.
| maximal subgroup of | #99933, #99938, #99949, … |
| maximal normal subgroup of | #99933, #99938, #99949, … |
| maximal quotient of | #219063, #219072, #219095, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15209);