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 | |
1530 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)(3,8)(4,5)(6,7)(11,15)(14,16)(25,26,30,33)(27,34,31,28)(29,37,38,32)(35,36,39,40);(1,3)(2,5)(4,6)(7,8)(17,18,21,20)(19,23,24,22)(27,31)(28,34)(35,39)(36,40);(2,7)(3,4)(9,10,13,12)(11,14,15,16)(17,19,21,24)(18,22,20,23)(25,27,30,31)(26,28,33,34)(29,35,38,39)(32,36,37,40);(1,4,6,3)(2,5,7,8)(9,11)(10,14)(12,16)(13,15)(25,28)(26,31)(27,33)(29,36)(30,34)(32,39)(35,37)(38,40);(1,5)(2,4)(3,7)(6,8)(9,12,13,10)(11,16,15,14)(17,20,21,18)(19,23,24,22)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,c^2=ghi,d^2=f,e^2=fhi,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, 3164386);