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 | #2539 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4394 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,26)(27,29)(28,36)(30,34)(31,33)(32,38)(35,39)(37,40);(1,2)(3,7)(4,8)(5,6)(25,27,30,33)(26,31,34,29)(28,35,38,40)(32,39,36,37);(1,3,5,4)(2,7,6,8)(10,14)(15,17)(16,21)(22,23)(25,26,30,34)(27,29,33,31)(28,36,38,32)(35,39,40,37);(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,39)(36,38)(37,40);(1,4,5,3)(2,8,6,7)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,28)(26,32)(27,35)(29,37)(30,38)(31,39)(33,40)(34,36);(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,29,30,31)(26,33,34,27)(28,37,38,39)(32,40,36,35)Polycyclic presentation:
2^9|b^2=i,c^2=g,e^2=gh,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, 7616949);