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 | #40452 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4360 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)(11,12)(13,21)(14,18)(15,16)(17,23)(19,24)(20,22)(25,26)(27,29)(28,36)(30,34)(31,33)(32,38)(35,39)(37,40);(1,2,4,5)(3,6,7,8)(9,11,15,14)(10,12,16,18)(13,19,23,22)(17,20,21,24)(25,27,30,33)(26,31,34,29)(28,35,38,40)(32,39,36,37);(10,16)(12,18)(17,21)(20,24)(25,26,30,34)(27,29,33,31)(28,36,38,32)(35,39,40,37);(1,3,4,7)(2,6,5,8)(9,12,15,18)(10,11,16,14)(13,20,23,24)(17,19,21,22)(25,26)(27,31)(28,32)(29,33)(30,34)(35,39)(36,38)(37,40);(9,13,15,23)(10,17,16,21)(11,19,14,22)(12,20,18,24)(25,28)(26,32)(27,35)(29,37)(30,38)(31,39)(33,40)(34,36);(9,14,15,11)(10,18,16,12)(13,22,23,19)(17,24,21,20)(25,29,30,31)(26,33,34,27)(28,37,38,39)(32,40,36,35)Polycyclic presentation:
2^9|b^2=ghi,c^2=i,d^2=gh,e^2=hi,f^2=h,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, 7618840);