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 | |
4568 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(11,17)(15,18)(19,23)(22,24)(25,26,31,29)(27,30,35,32)(28,33,37,38)(34,36,40,39);(2,6)(3,7)(9,10,13,14)(11,15,17,18)(12,16,20,21)(19,22,23,24)(25,27,31,35)(26,32,29,30)(28,34,37,40)(33,39,38,36);(1,3)(2,4)(5,7)(6,8)(10,14)(12,20)(15,18)(19,23)(26,29)(30,32)(33,38)(36,39);(1,4)(2,3)(5,8)(6,7)(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24)(25,28)(26,33)(27,34)(29,38)(30,36)(31,37)(32,39)(35,40);(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,29,31,26)(27,30,35,32)(28,33,37,38)(34,39,40,36);(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,30)(26,27)(28,36)(29,35)(31,32)(33,34)(37,39)(38,40)Polycyclic presentation:
2^9|a^2=g,b^2=gi,d^2=ghi,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=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, 8946647);