Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8736 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #381 |
| commutator | C24 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
481 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 481 conjugacy classes of subgroups — too many to draw. Only its 111 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:
(5,6,8,12,9,10,16,17)(7,13,15,20,14,11,18,19);(1,2)(3,4)(6,10)(8,16)(11,13)(15,18);(2,4)(5,7)(6,11,10,13)(8,15)(9,14)(12,19,17,20)(16,18)Polycyclic presentation:
2^8|a^2=g,c^2=d,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,g^b=gh,e^c=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #94484, #94485, #133428, … |
| maximal normal subgroup of | #94484, #94485, #133428, … |
| maximal quotient of | #118676, #118732, #118734, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4155);