Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
1546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,5)(3,6,7,8)(9,10,12,14)(11,13,15,16)(26,28)(29,37)(31,35)(33,39);(2,5)(3,7)(18,21)(22,24)(25,26,30,28)(27,31,34,35)(29,36,37,32)(33,40,39,38);(2,5)(6,8)(17,18,20,21)(19,22,23,24)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,40)(37,39);(1,2)(3,8)(4,5)(6,7)(9,10)(11,13)(12,14)(15,16)(17,19,20,23)(18,22,21,24)(25,28,30,26)(27,31,34,35)(29,32,37,36)(33,40,39,38);(1,3,4,7)(2,6,5,8)(9,11,12,15)(10,13,14,16)(25,29,30,37)(26,32,28,36)(27,33,34,39)(31,38,35,40)Polycyclic presentation:
2^9|a^2=fgi,b^2=fg,c^2=h,d^2=fgh,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3131668);