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 | |
4750 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,8,5,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,26,31,28)(27,32,34,29)(30,33,39,36)(35,40,38,37);(10,11)(13,21)(15,17)(20,24)(25,27)(26,32)(28,29)(30,38)(31,34)(33,37)(35,39)(36,40);(1,3,6,5)(2,7,4,8)(12,19)(15,17)(20,24)(22,23)(27,34)(29,32)(35,38)(37,40);(1,3,6,5)(2,8,4,7)(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23)(25,28,31,26)(27,32,34,29)(30,36,39,33)(35,40,38,37);(1,4,6,2)(3,8,5,7)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,29,31,32)(26,27,28,34)(30,37,39,40)(33,35,36,38);(1,5,6,3)(2,8,4,7)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,30,31,39)(26,33,28,36)(27,35,34,38)(29,37,32,40)Polycyclic presentation:
2^9|a^2=hi,c^2=h,d^2=ghi,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10314971);