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 | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10,14,15)(11,12,16,20)(13,21,22,17)(18,24,23,19)(25,26,30,34)(27,35,36,31)(28,32,33,29)(37,40,39,38);(1,3,4,8)(2,5,6,7)(10,15)(11,16)(17,21)(18,23)(27,36)(31,35)(37,39)(38,40);(9,11)(10,12)(13,18)(14,16)(15,20)(17,19)(21,24)(22,23)(25,27,30,36)(26,31,34,35)(28,39,33,37)(29,38,32,40);(25,28,30,33)(26,32,34,29)(27,37,36,39)(31,38,35,40);(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,21)(22,24)(25,26)(27,35)(28,32)(29,33)(30,34)(31,36)(37,40)(38,39);(9,13,14,22)(10,17,15,21)(11,18,16,23)(12,19,20,24)(25,29,30,32)(26,33,34,28)(27,38,36,40)(31,37,35,39)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=h,d^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7619542);