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 | #42984 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4413 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,15,17)(11,16,20,23)(12,14,19,24)(13,22,21,18)(25,26)(27,37)(28,34)(29,40)(30,36)(31,33)(32,39)(35,38);(1,2,4,5)(3,6,7,8)(9,11)(10,16)(12,21)(13,19)(14,18)(15,20)(17,23)(22,24)(25,27)(26,32)(28,35)(29,34)(30,38)(31,39)(33,37)(36,40);(10,17)(11,20)(12,19)(18,22)(26,33)(28,36)(32,37)(35,40);(9,12,15,19)(10,14,17,24)(11,21,20,13)(16,18,23,22)(25,28,31,36)(26,34,33,30)(27,35,39,40)(29,37,38,32);(9,13,15,21)(10,18,17,22)(11,19,20,12)(14,23,24,16)(25,29,31,38)(26,35,33,40)(27,34,39,30)(28,37,36,32);(1,3,4,7)(2,6,5,8)(9,14)(10,19)(11,18)(12,17)(13,16)(15,24)(20,22)(21,23)(25,30,31,34)(26,36,33,28)(27,38,39,29)(32,40,37,35)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7765099);