Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
925 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,15)(12,13)(14,16);(2,5)(3,8)(9,11,13,16)(10,14,12,15)(17,18)(19,23)(20,22)(21,24);(2,5)(6,7)(11,16)(14,15)(17,18,20,22)(19,21,24,23);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,14)(13,16)(17,18)(19,21)(20,22)(23,24);(1,3,4,8)(2,6,5,7)(9,12,13,10)(11,14,16,15)(17,19)(18,21)(20,24)(22,23)Polycyclic presentation:
2^8|a^2=g,b^2=fg,c^2=fgh,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | #132851, #486903, #487031, … |
| maximal normal subgroup of | #132851, #486903, #487031, … |
| maximal quotient of | #585261, #585335, #587553, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49356);