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 | |
1497 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)(3,7,8,6)(11,15)(14,16)(26,30)(27,33)(32,36)(35,37);(2,5)(6,7)(18,22)(23,24)(25,26,29,30)(27,34,33,31)(28,36,38,32)(35,39,37,40);(3,8)(6,7)(9,10,13,12)(11,14,15,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,38)(37,39);(1,3)(2,6)(4,8)(5,7)(9,11,13,15)(10,14,12,16)(17,19,21,20)(18,23,22,24)(25,27,29,33)(26,31,30,34)(28,37,38,35)(32,40,36,39);(1,2)(3,7)(4,5)(6,8)(9,12,13,10)(11,16,15,14)(17,20,21,19)(18,24,22,23)(25,28)(26,32)(27,35)(29,38)(30,36)(31,39)(33,37)(34,40)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=fi,d^2=fhi,e^2=fghi,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, 3232358);