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) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | #2619 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
789 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,7,8,6)(9,10,15,16)(11,12,13,14)(17,18)(19,22)(20,25)(21,26)(23,27)(24,29)(28,30)(31,32);(2,5)(3,8)(9,11,15,13)(10,12,16,14)(18,19)(20,27)(24,26)(28,32);(2,5)(6,7)(9,12,15,14)(10,11,16,13)(17,19,22,18)(20,25,27,23)(21,26,29,24)(28,31,32,30);(1,2)(3,6)(4,5)(7,8)(9,13,15,11)(10,12,16,14)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32);(1,3,4,8)(2,6,5,7)(9,14,15,12)(10,11,16,13)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=gh,b^2=fg,c^2=fgh,d^2=fg,e^2=g,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 | #132848, #486873, #487001, … |
| maximal normal subgroup of | #132848, #486873, #487001, … |
| maximal quotient of | #585261, #586407, #587541, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50118);