Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C23 | C22×C8 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
579 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 579 conjugacy classes of subgroups — too many to draw. Only its 107 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 — 3 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,15,16,13)(12,14)(17,18,19,21,20,22,23,24);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(9,11)(10,13)(12,16)(14,15)Polycyclic presentation:
2^8|a^2=f,c^2=d,f^2=h,b^a=bd,c^a=ce,e^a=eg,f^c=fg64 conjugacy classes — view the full character table.
| maximal subgroup of | #93300, #93557, #97065, … |
| maximal normal subgroup of | #93300, #93557, #97065, … |
| maximal quotient of | #99096, #99097, #99098, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1686);