Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11508 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C22×D4 | C23 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
393 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 393 conjugacy classes of subgroups — too many to draw. Only its 59 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16,20,15)(13,17)(18,22)(19,21,24,23);(2,5)(6,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,21)(17,22)(20,24);(1,3,4,7)(2,6,5,8)(10,15)(11,18)(12,20)(14,21)(16,17)(22,23)Polycyclic presentation:
2^8|a^2=f,c^2=d,e^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gh,c^b=ch,e^c=eh,f^c=fgh,f^e=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #133498, #133499, #133500, … |
| maximal normal subgroup of | #133498, #133499, #133500, … |
| maximal quotient of | #135167, #135172, #135209, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5846);