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 | |
4661 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,15,18)(11,19,22,16)(12,21,23,13)(14,17,24,20)(25,26)(27,34)(28,32)(29,31)(30,33)(35,40)(36,39)(37,38);(1,2,4,6)(3,5,8,7)(9,11,15,22)(10,16,18,19)(12,20,23,17)(13,14,21,24)(28,36)(32,39)(35,37)(38,40);(1,2)(3,5)(4,6)(7,8)(25,27,30,29)(26,31,33,34)(28,35,36,37)(32,38,39,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,13)(11,20)(14,16)(15,23)(17,22)(18,21)(19,24)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,39)(34,40);(9,13)(10,12)(11,14)(15,21)(16,20)(17,19)(18,23)(22,24)(25,29,30,27)(26,31,33,34)(28,37,36,35)(32,38,39,40);(9,14)(10,17)(11,21)(12,16)(13,22)(15,24)(18,20)(19,23)(25,26)(27,31)(28,32)(29,34)(30,33)(35,38)(36,39)(37,40)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8004220);