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 | #42299 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4194 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,15)(12,19)(13,16)(14,17)(18,23)(20,21)(22,24)(25,26)(27,34)(28,31)(29,36)(30,33)(32,38)(35,37)(39,40);(1,2)(3,7)(4,5)(6,8)(9,10,14,17)(11,15,19,12)(13,16,22,24)(18,23,21,20)(26,28)(27,33)(32,36)(35,39);(2,5)(6,7)(9,11,14,19)(10,15,17,12)(13,21,22,18)(16,20,24,23)(25,27)(26,30)(28,34)(29,35)(31,33)(32,37)(36,40)(38,39);(3,8)(6,7)(9,12)(10,11)(13,20)(14,15)(16,18)(17,19)(21,24)(22,23)(25,28,31,26)(27,34,33,30)(29,36,38,32)(35,40,39,37);(1,3,4,8)(2,6,5,7)(9,10)(11,15)(12,19)(13,16)(14,17)(18,23)(20,21)(22,24)(25,29)(26,32)(27,35)(28,36)(30,37)(31,38)(33,39)(34,40);(1,2)(3,6)(4,5)(7,8)(9,13,14,22)(10,16,17,24)(11,18,19,21)(12,20,15,23)(25,30,31,34)(26,33,28,27)(29,37,38,40)(32,39,36,35)Polycyclic presentation:
2^9|b^2=gh,c^2=gh,d^2=h,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8269374);