Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1563 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,12)(11,15,16,14)(25,26)(27,33)(28,30)(29,36)(31,35)(32,37)(34,38)(39,40);(1,2,4,6)(3,5,8,7)(18,22)(23,24)(26,28)(29,37)(31,33)(34,40);(3,8)(5,7)(9,10,13,12)(11,14,16,15)(17,18,21,22)(19,23,20,24)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(1,3)(2,5)(4,8)(6,7)(9,11,13,16)(10,14,12,15)(17,19,21,20)(18,23,22,24)(25,28,30,26)(27,31,35,33)(29,36,37,32)(34,38,40,39);(1,2)(3,5)(4,6)(7,8)(9,12,13,10)(11,15,16,14)(17,20,21,19)(18,24,22,23)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=fi,b^2=fg,c^2=fghi,d^2=fghi,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1758966);