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 | |
4365 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,19)(12,14)(13,16)(15,18)(17,24)(20,21)(22,23)(25,26,30,34)(27,35,36,31)(28,32,29,33)(37,40,38,39);(1,2,4,5)(3,6,7,8)(9,11,15,13)(10,16,18,19)(12,22,17,20)(14,21,24,23)(27,36)(31,35)(37,38)(39,40);(11,13)(16,19)(20,22)(21,23)(25,27)(26,31)(28,37)(29,38)(30,36)(32,39)(33,40)(34,35);(1,3,4,7)(2,6,5,8)(9,12,15,17)(10,14,18,24)(11,20,13,22)(16,23,19,21)(25,26)(27,31)(28,32)(29,33)(30,34)(35,36)(37,39)(38,40);(9,13,15,11)(10,16,18,19)(12,22,17,20)(14,23,24,21)(25,28,30,29)(26,32,34,33)(27,37,36,38)(31,39,35,40);(9,14)(10,17)(11,21)(12,18)(13,23)(15,24)(16,20)(19,22)(25,29,30,28)(26,33,34,32)(27,38,36,37)(31,40,35,39)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=g,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,d^b=dhi,e^c=ehi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7551386);