Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1878 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)(11,13)(12,15)(18,21)(19,23)(25,26)(27,28)(29,37)(30,31)(32,34)(33,39)(35,38)(36,40);(9,10,14,16)(11,15,13,12)(18,21)(22,24)(26,31)(27,34)(33,37)(35,40);(9,11,14,13)(10,15,16,12)(17,18,20,21)(19,22,23,24)(25,27)(26,32)(28,31)(29,35)(30,34)(33,38)(36,37)(39,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,16)(14,15)(25,28,30,32)(26,27,31,34)(29,38,39,36)(33,40,37,35);(1,3,4,7)(2,6,5,8)(9,13,14,11)(10,12,16,15)(17,19,20,23)(18,22,21,24)(25,29,30,39)(26,33,31,37)(27,35,34,40)(28,36,32,38)Polycyclic presentation:
2^9|b^2=gh,c^2=fg,d^2=f,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1245081);