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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1436 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,4)(3,8,7,5)(9,10,13,12)(11,14,15,16)(17,18,22,24)(19,20,23,21)(25,26)(27,28)(29,32)(30,31)(33,36)(34,35)(37,38)(39,40);(11,15)(14,16)(17,19,22,23)(18,20,24,21)(26,31)(27,33)(29,37)(35,40);(1,3)(2,5)(4,8)(6,7)(9,11,13,15)(10,14,12,16)(17,18)(19,20)(21,23)(22,24)(25,27,30,33)(26,28,31,36)(29,34,37,39)(32,35,38,40);(1,4,6,2)(3,8,7,5)(17,20,22,21)(18,19,24,23)(25,28)(26,27)(29,35)(30,36)(31,33)(32,34)(37,40)(38,39);(1,5,6,8)(2,7,4,3)(9,12,13,10)(11,16,15,14)(17,21,22,20)(18,23,24,19)(25,29,30,37)(26,32,31,38)(27,34,33,39)(28,35,36,40)Polycyclic presentation:
2^9|a^2=fgh,b^2=i,c^2=h,d^2=g,e^2=gh,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, 1033862);