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 | |
4479 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)(11,18)(12,15)(13,16)(14,17)(19,24)(20,23)(21,22)(25,26)(27,32)(28,37)(29,34)(30,38)(31,33)(35,39)(36,40);(1,2,4,6)(3,5,8,7)(9,11,14,12)(10,15,17,18)(13,19,22,20)(16,23,21,24)(25,27,31,35)(26,32,33,39)(28,36,38,29)(30,34,37,40);(1,2)(3,5)(4,6)(7,8)(26,33)(27,35)(28,38)(34,40);(1,3,4,8)(2,5,6,7)(9,12,14,11)(10,15,17,18)(13,20,22,19)(16,23,21,24)(25,28)(26,30)(27,36)(29,35)(31,38)(32,34)(33,37)(39,40);(9,10)(11,18)(12,15)(13,21)(14,17)(16,22)(19,23)(20,24)(25,29,31,36)(26,34,33,40)(27,28,35,38)(30,32,37,39);(9,13)(10,16)(11,19)(12,20)(14,22)(15,23)(17,21)(18,24)(25,30)(26,28)(27,34)(29,39)(31,37)(32,36)(33,38)(35,40)Polycyclic presentation:
2^9|b^2=gh,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9818014);