Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10390 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #393 |
| commutator, Frattini, cosocle | C25 | C23 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
373 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 373 conjugacy classes of subgroups — too many to draw. Only its 45 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)(3,5)(4,6)(7,8)(9,10,13,16)(11,14,18,23)(12,19)(15,21)(17,24)(20,22);(1,3)(2,5)(4,7)(6,8)(9,11)(10,14)(12,20,21,17)(13,18)(15,22,19,24)(16,23);(5,8)(9,12)(10,15)(11,17)(13,21)(14,22)(16,19)(18,20)(23,24)Polycyclic presentation:
2^8|a^2=dg,b^2=d,b^a=bd,c^a=ce,e^a=eg,f^a=fgh,c^b=cf,e^b=eh,f^b=fg,d^c=dg28 conjugacy classes — view the full character table.
| maximal subgroup of | #94701, #94702, #151367, … |
| maximal normal subgroup of | #94701, #94702, #151367, … |
| maximal quotient of | #104757, #104758, #104761, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3380);