Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| automorphism | Aut(G) order 3145728 |
| 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 | |
1814 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(9,10,14,15)(11,12,13,16)(18,22)(19,24)(25,26,30,28)(27,33,34,31)(29,36,37,32)(35,38,40,39);(9,11,14,13)(10,12,15,16)(17,18)(19,20)(21,22)(23,24)(26,28)(27,34)(32,36)(35,40);(10,15)(12,16)(17,19,21,24)(18,20,22,23)(25,27,30,34)(26,31,28,33)(29,35,37,40)(32,38,36,39);(1,2)(3,6)(4,5)(7,8)(9,12,14,16)(10,11,15,13)(17,19)(18,23)(20,22)(21,24)(25,28,30,26)(27,31,34,33)(29,36,37,32)(35,38,40,39);(1,3,4,7)(2,6,5,8)(9,13,14,11)(10,16,15,12)(17,20,21,23)(18,19,22,24)(25,29,30,37)(26,32,28,36)(27,35,34,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=fgh,b^2=fg,c^2=f,d^2=fgh,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,d^c=dg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4839583);