Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #17082 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
691 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 691 conjugacy classes of subgroups — too many to draw. Only its 143 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,5)(6,8)(9,10,12,14)(11,13,15,16)(17,18,20,21)(19,22,24,23)(26,27)(30,31);(2,5)(3,7)(11,15)(13,16)(18,21)(22,23)(25,26,29,27)(28,30,32,31);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19,20,24)(18,22,21,23)(25,27,29,26)(28,30,32,31);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,23)(20,21)(22,24)(25,28)(26,30)(27,31)(29,32)Polycyclic presentation:
2^8|a^2=efg,b^2=eh,c^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,d^c=de46 conjugacy classes — view the full character table.
| maximal subgroup of | #125851, #126172, #126227, … |
| maximal normal subgroup of | #125851, #126172, #126227, … |
| maximal quotient of | #156505, #156509, #156514, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10222);