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 | |
4632 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)(4,5)(6,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,26)(27,31)(28,37)(29,36)(30,32)(33,38)(34,40)(35,39);(2,5)(6,7)(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21)(25,27,30,35)(26,31,32,39)(28,36,38,34)(29,33,40,37);(1,2)(3,6)(4,5)(7,8)(12,20)(15,22)(18,24)(21,23)(26,32)(27,35)(33,37)(34,36);(1,3,4,8)(2,6,5,7)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,28)(26,33)(27,36)(29,39)(30,38)(31,40)(32,37)(34,35);(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,29,30,40)(26,34,32,36)(27,37,35,33)(28,31,38,39);(9,12)(10,15)(11,18)(13,20)(14,21)(16,22)(17,23)(19,24)(25,26)(27,31)(28,33)(29,34)(30,32)(35,39)(36,40)(37,38)Polycyclic presentation:
2^9|b^2=gi,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, 8314307);