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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4498 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)(11,17)(12,20)(13,14)(15,18)(16,21)(19,22)(23,24)(25,26)(27,33)(28,30)(29,34)(31,32)(35,40)(36,37)(38,39);(1,2)(3,6)(4,5)(7,8)(11,18)(15,17)(19,24)(22,23)(26,32)(27,28)(34,39)(35,36);(1,3,4,7)(2,6,5,8)(10,14)(15,17)(16,20)(22,23)(25,27,31,28)(26,33,32,30)(29,36,38,35)(34,37,39,40);(9,11,13,18)(10,15,14,17)(12,19,21,24)(16,22,20,23)(25,28,31,27)(26,30,32,33)(29,36,38,35)(34,37,39,40);(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,29,31,38)(26,34,32,39)(27,35,28,36)(30,37,33,40);(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(25,30)(26,27)(28,32)(29,37)(31,33)(34,35)(36,39)(38,40)Polycyclic presentation:
2^9|c^2=g,d^2=hi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,d^b=dhi,e^c=ehi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7551536);