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 | #2847 | C22 |
| 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) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,15,18)(11,19,22,13)(12,21,24,16)(14,20,23,17)(28,35)(32,37)(34,39)(38,40);(1,2,4,6)(3,5,8,7)(9,11,15,22)(10,13,18,19)(12,17,24,20)(14,21,23,16)(26,30)(27,33)(32,37)(34,39);(1,2)(3,5)(4,6)(7,8)(9,12,15,24)(10,16,18,21)(11,20,22,17)(13,23,19,14)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,20)(14,21)(15,22)(16,23)(17,24)(18,19);(9,13)(10,11)(12,23)(14,24)(15,19)(16,20)(17,21)(18,22)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(9,14,15,23)(10,17,18,20)(11,21,22,16)(12,13,24,19)(25,28)(26,32)(27,34)(29,35)(30,37)(31,38)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000577);