Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1393169 |
| 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 | |
1724 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,8)(3,4,7,5)(17,18,20,22)(19,23,24,21)(25,26,29,30)(27,31,35,33)(28,36,38,32)(34,40,37,39);(1,3,6,7)(2,4,8,5)(10,13)(14,16)(19,24)(21,23)(26,30)(31,33)(32,36)(39,40);(1,4,6,5)(2,3,8,7)(9,10,12,13)(11,14,15,16)(17,19)(18,21)(20,24)(22,23)(25,26)(27,33)(28,32)(29,30)(31,35)(34,40)(36,38)(37,39);(1,2)(3,4)(5,7)(6,8)(25,27,29,35)(26,31,30,33)(28,37,38,34)(32,40,36,39);(1,5,6,4)(2,7,8,3)(9,11,12,15)(10,14,13,16)(17,18)(19,21)(20,22)(23,24)(25,28,29,38)(26,32,30,36)(27,34,35,37)(31,39,33,40)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gh,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^c=df,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1309693);