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 | |
4387 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)(11,14)(12,19)(13,16)(15,20)(17,21)(18,24)(22,23)(25,26,30,33)(27,29,36,39)(28,32,38,31)(34,40,35,37);(1,2,4,6)(3,5,8,7)(11,17)(14,21)(18,23)(22,24)(25,27,30,36)(26,29,33,39)(28,35,38,34)(31,40,32,37);(1,2)(3,5)(4,6)(7,8)(9,10,13,16)(11,14,17,21)(12,15,20,19)(18,22,23,24)(25,28,30,38)(26,31,33,32)(27,34,36,35)(29,37,39,40);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,20)(23,24)(25,28)(26,32)(27,35)(29,37)(30,38)(31,33)(34,36)(39,40);(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,27)(26,29)(28,35)(30,36)(31,40)(32,37)(33,39)(34,38);(9,12)(10,15)(11,18)(13,20)(14,22)(16,19)(17,23)(21,24)(25,29)(26,27)(28,37)(30,39)(31,34)(32,35)(33,36)(38,40)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=h,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9150403);