Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C8 | C24 |
| commutator, socle | C22 | C23×C8 |
| prime radical | C22×D4 | C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
309 conjugacy classes of subgroups; only the 187 normal subgroups are shown.
This group has 309 conjugacy classes of subgroups — too many to draw. Only its 187 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) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,11,15,12,16,17,22,13,14,18,20,19,21,23,24);(1,3)(2,5)(4,8)(6,7);(3,8)(5,7)(10,14)(15,20)(16,21)(22,24);(2,6)(5,7)Polycyclic presentation:
2^8|a^2=f,f^2=g,g^2=h,b^a=be,c^a=ch,d^a=de,c^b=ce88 conjugacy classes — view the full character table.
| maximal subgroup of | #135857, #135935, #135936, … |
| maximal normal subgroup of | #135857, #135935, #135936, … |
| maximal quotient of | #493264, #493265, #493277, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26475);