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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4593 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,26)(27,32)(28,34)(29,37)(30,36)(31,33)(35,40)(38,39);(1,2,5,4)(3,7,8,6)(10,14)(15,17)(16,21)(22,23)(25,27,31,30)(26,32,33,36)(28,37,38,35)(29,39,40,34);(2,4)(6,7)(12,20)(16,21)(18,24)(22,23)(26,33)(27,30)(34,39)(35,37);(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,28)(26,34)(27,35)(29,36)(30,37)(31,38)(32,40)(33,39);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,29)(26,35)(27,34)(28,36)(30,39)(31,40)(32,38)(33,37);(1,4,5,2)(3,7,8,6)(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,30,31,27)(26,36,33,32)(28,37,38,35)(29,39,40,34)Polycyclic presentation:
2^9|b^2=i,d^2=g,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8161561);