Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) order 32768 |
| 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, prime radical | C23 | C23×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
341 conjugacy classes of subgroups; only the 215 normal subgroups are shown.
This group has 341 conjugacy classes of subgroups — too many to draw. Only its 215 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) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(9,10)(11,14)(12,16)(13,15)(17,18,20,23,21,19,22,24);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,15,16)(17,19,20,24,21,18,22,23);(2,5)(6,8)(9,12,13,11)(10,14,15,16);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^8|a^2=g,b^2=fg,c^2=f,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=ce88 conjugacy classes — view the full character table.
| maximal subgroup of | #212324, #212325, #212326, … |
| maximal normal subgroup of | #212324, #212325, #212326, … |
| maximal quotient of | #355244, #355382, #355866, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15287);