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 | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
828 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(4,8)(10,13)(11,15)(17,18)(19,24)(20,25)(21,29)(22,23)(26,30)(27,28)(31,32);(1,2,5,6)(3,8,7,4)(9,10,12,13)(11,16,15,14)(18,23)(20,24)(26,29)(28,31);(2,6)(3,7)(9,10,12,13)(11,14,15,16)(17,19,22,25)(18,24,23,20)(21,27,30,32)(26,31,29,28);(1,3,5,7)(2,4,6,8)(9,10)(11,14)(12,13)(15,16)(17,20,22,24)(18,25,23,19)(21,28,30,31)(26,32,29,27);(1,4)(2,7)(3,6)(5,8)(9,11)(10,14)(12,15)(13,16)(17,21,22,30)(18,26,23,29)(19,27,25,32)(20,28,24,31)Polycyclic presentation:
2^8|b^2=gh,c^2=fh,d^2=fg,e^2=fgh,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 | #133068, #487064, #487192, … |
| maximal normal subgroup of | #133068, #487064, #487192, … |
| maximal quotient of | #585276, #586050, #587382, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51972);