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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1758 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(10,13)(11,15)(17,18,20,22)(19,23,24,21)(25,26,30,31)(27,35,36,32)(28,33,37,40)(29,38,39,34);(1,2,5,4)(3,6,7,8)(10,13)(14,16)(26,31)(27,36)(33,40)(34,38);(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24)(25,27,30,36)(26,32,31,35)(28,34,37,38)(29,33,39,40);(1,3)(2,6)(4,8)(5,7)(25,28)(26,33)(27,34)(29,35)(30,37)(31,40)(32,39)(36,38);(1,4,5,2)(3,8,7,6)(9,11,12,15)(10,14,13,16)(17,19,20,24)(18,21,22,23)(25,29,30,39)(26,34,31,38)(27,33,36,40)(28,32,37,35)Polycyclic presentation:
2^9|a^2=fgh,b^2=i,c^2=f,e^2=gi,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, 1252658);