Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C23 | C2×C22⋊C4 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
248 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 81 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 — 3 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(10,13)(11,14)(15,16)(17,18,22,23,21,20,19,24);(2,5)(6,8)(9,10,12,13)(11,14,15,16)(17,19,21,22)(18,23,20,24);(1,3,4,7)(2,6,5,8)(9,11,12,15)(10,14,13,16)(17,20,21,18)(19,23,22,24)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=d,e^2=g,f^2=h,b^a=bd,c^a=ce,e^a=eg,c^b=ch,e^c=eg64 conjugacy classes — view the full character table.
| maximal subgroup of | #97783, #97785, #97787, … |
| maximal normal subgroup of | #97783, #97785, #97787, … |
| maximal quotient of | #99110, #99111, #99112, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1699);