Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator | C22 | C23×C8 |
| socle | C23 | C23×C4 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
348 conjugacy classes of subgroups; only the 193 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 193 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 — 4 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(9,10,12,14)(11,13,16,15)(17,18,19,21,20,22,23,24);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16);(2,5)(6,7)(9,11,12,16)(10,13,14,15);(1,3,4,8)(2,6,5,7)(10,14)(13,15)Polycyclic presentation:
2^8|a^2=g,c^2=e,d^2=f,g^2=h,b^a=be,c^a=cf,d^a=def,c^b=cef,d^b=de76 conjugacy classes — view the full character table.
| Fitting subgroup of | (#126⋊C3)⋊C8 |
| maximal subgroup of | C2×#23375, #6407738, #6407740, … |
| maximal normal subgroup of | C2×#23375, #6407738, #6407740, … |
| maximal quotient of | #357312, #357332, #357389, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16363);