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 | |
4854 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,26)(27,32)(28,37)(29,33)(30,31)(34,35)(36,39)(38,40);(1,2,4,6)(3,5,8,7)(10,14)(11,17)(16,21)(19,23)(26,31)(28,33)(32,39)(34,38);(1,2)(3,5)(4,6)(7,8)(12,20)(16,21)(19,23)(22,24)(26,31)(29,37)(32,39)(35,40);(1,3,4,8)(2,5,6,7)(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24)(25,27,30,36)(26,32,31,39)(28,38,33,34)(29,35,37,40);(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,28,30,33)(26,29,31,37)(27,34,36,38)(32,40,39,35);(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,29)(26,33)(27,35)(28,31)(30,37)(32,34)(36,40)(38,39)Polycyclic presentation:
2^9|b^2=i,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8314305);