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 | |
4524 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)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,26)(27,32)(28,37)(29,34)(30,38)(31,33)(35,39)(36,40);(2,5)(6,7)(10,14)(15,17)(16,21)(22,23)(25,27,31,35)(26,32,33,39)(28,34,38,40)(29,30,36,37);(1,2)(3,6)(4,5)(7,8)(12,20)(16,21)(18,24)(22,23)(26,33)(27,35)(30,37)(34,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,28)(26,30)(27,34)(29,39)(31,38)(32,36)(33,37)(35,40);(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,29,31,36)(26,34,33,40)(27,30,35,37)(28,32,38,39);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,30)(26,28)(27,36)(29,35)(31,37)(32,34)(33,38)(39,40)Polycyclic presentation:
2^9|b^2=h,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8832702);