Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #117 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #383 |
| prime radical | C24 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
337 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 337 conjugacy classes of subgroups — too many to draw. Only its 135 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) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6)(7,15,18,19)(8,11)(9,13)(10,14)(12,17,20,16);(6,11,14,13)(12,19,20,15);(2,4)(5,7,8,16,10,18,9,17)(6,12,13,15,14,20,11,19)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=d,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #93360, #128804, #128805, … |
| maximal normal subgroup of | #93360, #128804, #128805, … |
| maximal quotient of | #108490, #108491, #108494, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3801);