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 | |
4465 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(12,20)(16,21)(18,24)(22,23)(25,26,30,32)(27,35,36,31)(28,33,38,40)(29,39,37,34);(1,2)(3,5)(4,6)(7,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,27,30,36)(26,31,32,35)(28,29,38,37)(33,34,40,39);(10,14)(15,17)(16,21)(22,23)(26,32)(27,36)(28,38)(34,39);(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,28,30,38)(26,33,32,40)(27,37,36,29)(31,39,35,34);(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,26)(27,31)(28,33)(29,34)(30,32)(35,36)(37,39)(38,40);(1,3,4,7)(2,5,6,8)(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,29,30,37)(26,34,32,39)(27,38,36,28)(31,40,35,33)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=i,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7621092);