Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #121 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
341 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 341 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(5,6,7,8)(10,14)(11,18)(12,15)(16,23)(17,21)(20,24);(1,2)(3,4)(9,10,15,21,13,17,12,14)(11,16,22,23,18,24,19,20);(2,4)(9,11,12,19,13,18,15,22)(10,16,14,20,17,24,21,23)Polycyclic presentation:
2^8|a^2=g,b^2=dh,c^2=d,d^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,e^c=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #131362, #131363, #131364, … |
| maximal normal subgroup of | #131362, #131363, #131364, … |
| maximal quotient of | #120171, #120191, #120194, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4227);