Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2846 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle, prime radical | C23 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
411 conjugacy classes of subgroups; only the 223 normal subgroups are shown.
This group has 411 conjugacy classes of subgroups — too many to draw. Only its 223 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:
(2,4)(5,6)(7,12)(8,10)(9,11)(13,14,16,15,17,18,20,19);(5,7,9,8)(6,10,11,12);(1,2,3,4);(5,8,9,7)(6,10,11,12)(13,15,16,18,17,19,20,14)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=f,d^2=eg,g^2=h,b^a=be,c^a=cf,d^a=dh88 conjugacy classes — view the full character table.
| maximal subgroup of | #96871, #96872, #96873, … |
| maximal normal subgroup of | #96871, #96872, #96873, … |
| maximal quotient of | #213370, #213371, #213372, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14700);