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 | |
4544 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)(11,15)(12,14)(13,16)(17,19)(18,22)(20,21)(23,24)(25,26)(27,36)(28,34)(29,31)(30,37)(32,38)(33,39)(35,40);(1,2,4,6)(3,5,8,7)(10,12)(11,17)(16,20)(18,23)(25,27)(26,32)(28,35)(29,36)(30,34)(31,38)(33,40)(37,39);(1,2)(3,5)(4,6)(7,8)(11,17)(13,21)(15,19)(16,20)(25,28,31,39)(26,33,29,34)(27,35,38,37)(30,32,40,36);(1,3,4,8)(2,5,6,7)(9,11,14,17)(10,15,12,19)(13,18,21,23)(16,22,20,24)(25,28)(26,34)(27,37)(29,33)(30,36)(31,39)(32,40)(35,38);(9,12,14,10)(11,15,17,19)(13,20,21,16)(18,22,23,24)(25,29,31,26)(27,36,38,32)(28,34,39,33)(30,37,40,35);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,30,31,40)(26,35,29,37)(27,34,38,33)(28,36,39,32)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8842084);