Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C24 |
| commutator | C23 | C23×C4 |
| prime radical | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
749 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6,8,7)(9,10)(11,15)(12,13)(14,16)(17,18,20,22)(19,23,24,21);(2,4)(10,13)(11,16)(17,18,20,22)(19,21,24,23);(1,2,3,4)(5,7,8,6)(9,11)(10,14)(12,16)(13,15)(17,18)(19,21)(20,22)(23,24);(9,10)(11,14)(12,13)(15,16)(17,19)(18,21)(20,24)(22,23)Polycyclic presentation:
2^8|a^2=h,b^2=ef,c^2=egh,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de64 conjugacy classes — view the full character table.
| maximal subgroup of | #136185, #136261, #137316, … |
| maximal normal subgroup of | #136185, #136261, #137316, … |
| maximal quotient of | #153977, #153989, #154007, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7565);