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 | #51809 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,15,17)(11,16,12,14)(13,18,22,24)(19,23,20,21)(25,26,31,35)(27,33,29,32)(28,30,34,40)(36,37,38,39);(1,2,4,5)(3,6,7,8)(9,11,15,12)(10,16,17,14)(13,19,22,20)(18,23,24,21)(25,27,31,29)(26,32,35,33)(28,38,34,36)(30,37,40,39);(10,17)(14,16)(18,24)(21,23)(27,29)(32,33)(36,38)(37,39);(9,12,15,11)(10,14,17,16)(13,19,22,20)(18,23,24,21)(25,28)(26,30)(27,36)(29,38)(31,34)(32,39)(33,37)(35,40);(1,3,4,7)(2,6,5,8)(9,13)(10,18)(11,19)(12,20)(14,21)(15,22)(16,23)(17,24)(25,29,31,27)(26,33,35,32)(28,36,34,38)(30,39,40,37);(9,14)(10,11)(12,17)(13,21)(15,16)(18,19)(20,24)(22,23)(25,30)(26,34)(27,37)(28,35)(29,39)(31,40)(32,36)(33,38)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,d^2=h,e^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7988598);