Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C23 |
| commutator | C22 | C23×C8 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C23×C4 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
429 conjugacy classes of subgroups; only the 231 normal subgroups are shown.
This group has 429 conjugacy classes of subgroups — too many to draw. Only its 231 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,19,21,24,20,23);(2,5)(6,8)(17,19,22,24,21,23,20,18);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20,21,22)(18,23,24,19);(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=e,d^2=f,g^2=h,b^a=be,c^a=cf112 conjugacy classes — view the full character table.
| maximal subgroup of | #183724, #183725, #183728, … |
| maximal normal subgroup of | #183724, #183725, #183728, … |
| maximal quotient of | #353839, #353986, #354126, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14418);