Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4231 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,14,13)(11,15,19,18)(12,16,22,21)(17,23,20,24)(26,30)(31,34)(32,36)(38,40);(1,2)(3,6)(4,5)(7,8)(25,26)(27,31)(28,32)(29,30)(33,34)(35,38)(36,37)(39,40);(1,3,4,7)(2,6,5,8)(10,13)(15,18)(16,21)(23,24)(27,33)(31,34)(35,39)(38,40);(9,11,14,19)(10,15,13,18)(12,20,22,17)(16,24,21,23)(25,26)(27,34)(28,36)(29,30)(31,33)(32,37)(35,38)(39,40);(9,12,14,22)(10,16,13,21)(11,17,19,20)(15,23,18,24)(25,27,29,33)(26,31,30,34)(28,35,37,39)(32,38,36,40);(9,13,14,10)(11,18,19,15)(12,21,22,16)(17,24,20,23)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=h,c^2=gi,d^2=h,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7999219);