Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| 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 | |
1754 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,4)(3,8,5,7)(10,13)(11,15)(17,18,20,22)(19,23,24,21)(25,26)(27,32)(28,34)(29,33)(30,31)(35,39)(36,40)(37,38);(1,2,6,4)(3,7,5,8)(10,13)(14,16)(19,24)(21,23)(26,31)(27,34)(29,37)(36,39);(1,3,6,5)(2,7,4,8)(9,10)(11,14)(12,13)(15,16)(17,19,20,24)(18,21,22,23)(25,27,30,34)(26,32,31,28)(29,35,37,40)(33,39,38,36);(1,4,6,2)(3,8,5,7)(17,18)(19,21)(20,22)(23,24)(25,28,30,32)(26,27,31,34)(29,36,37,39)(33,35,38,40);(1,5,6,3)(2,8,4,7)(9,11)(10,14)(12,15)(13,16)(17,18)(19,21)(20,22)(23,24)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=fgh,c^2=fghi,d^2=gh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1099635);