Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
832 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,6)(4,7)(17,18,20,22)(19,23,24,21);(1,2)(3,7)(4,8)(5,6)(11,15)(13,16)(19,24)(21,23);(1,3,5,8)(2,7,6,4)(9,10,12,14)(11,13,15,16)(17,19,20,24)(18,21,22,23);(9,11,12,15)(10,13,14,16)(17,18)(19,21)(20,22)(23,24);(1,4,5,7)(2,8,6,3)Polycyclic presentation:
2^8|a^2=g,c^2=fgh,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #130365, #133717, #486044, … |
| maximal normal subgroup of | #130365, #133717, #486044, … |
| maximal quotient of | #532122, #532250, #532608, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52939);