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 | |
1682 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)(3,5)(4,7)(6,8)(9,10,12,14)(11,13,15,16)(17,18)(19,23)(20,21)(22,24)(25,26,29,32)(27,33,35,30)(28,36,37,31)(34,38,40,39);(2,3)(4,8)(18,20)(19,24)(25,26,29,32)(27,33,35,30)(28,31,37,36)(34,39,40,38);(2,3)(6,7)(17,18,21,20)(19,23,24,22)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,37)(39,40);(1,3,5,2)(4,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,19,21,24)(18,22,20,23)(25,27,29,35)(26,30,32,33)(28,34,37,40)(31,38,36,39);(1,4)(2,6)(3,7)(5,8)(9,11,12,15)(10,13,14,16)(17,20,21,18)(19,23,24,22)(25,28)(26,31)(27,34)(29,37)(30,38)(32,36)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=fi,b^2=fg,c^2=fgh,d^2=g,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, 1756710);