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 | #50170 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4363 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)(11,14,13,16)(12,20,21,18)(19,24,22,23)(26,31)(27,33)(32,37)(34,40);(1,2,4,5)(3,6,7,8)(9,11,15,13)(10,16,17,14)(12,19,21,22)(18,24,20,23)(25,26)(27,28)(29,37)(30,31)(32,39)(33,35)(34,38)(36,40);(10,17)(11,13)(12,21)(23,24)(26,31)(28,35)(32,37)(36,38);(9,12,15,21)(10,18,17,20)(11,19,13,22)(14,23,16,24)(25,27)(26,28)(29,34)(30,33)(31,35)(32,36)(37,38)(39,40);(9,13,15,11)(10,14,17,16)(12,22,21,19)(18,23,20,24)(25,28)(26,27)(29,38)(30,35)(31,33)(32,40)(34,37)(36,39);(1,3,4,7)(2,6,5,8)(9,14)(10,13)(11,17)(12,23)(15,16)(18,22)(19,20)(21,24)(25,29,30,39)(26,32,31,37)(27,34,33,40)(28,36,35,38)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7878869);