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 | |
4517 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)(3,4)(5,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,22)(18,21)(19,20)(23,24)(25,26)(27,32)(28,30)(29,35)(31,33)(34,36)(37,40)(38,39);(2,6)(3,7)(10,14)(15,21)(16,19)(22,23)(25,27)(26,32)(28,38)(29,36)(30,39)(31,37)(33,40)(34,35);(1,3,5,7)(2,4,6,8)(26,33)(27,37)(28,36)(35,39);(1,4)(2,7)(3,6)(5,8)(9,11,13,18)(10,15,14,21)(12,17,20,24)(16,22,19,23)(25,28,31,36)(26,34,33,30)(27,38,37,29)(32,35,40,39);(9,10)(11,15)(12,19)(13,14)(16,20)(17,23)(18,21)(22,24)(25,29,31,38)(26,35,33,39)(27,36,37,28)(30,32,34,40);(9,12)(10,16)(11,17)(13,20)(14,19)(15,22)(18,24)(21,23)(25,30,31,34)(26,36,33,28)(27,39,37,35)(29,40,38,32)Polycyclic presentation:
2^9|c^2=hi,d^2=gi,e^2=h,f^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, 9831119);