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 | |
4453 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,18)(12,13)(14,15)(16,21)(17,22)(19,24)(20,23)(26,31)(29,33)(32,38)(36,39);(1,2,5,6)(3,7,4,8)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,40)(37,38);(10,15)(12,17)(16,18)(19,23)(26,31)(28,34)(32,38)(35,40);(1,3,5,4)(2,7,6,8)(9,11,14,21)(10,16,15,18)(12,19,17,23)(13,20,22,24)(25,27,30,37)(26,32,31,38)(28,35,34,40)(29,36,33,39);(1,3,5,4)(2,7,6,8)(9,12,14,17)(10,13,15,22)(11,19,21,23)(16,20,18,24)(25,28,30,34)(26,33,31,29)(27,35,37,40)(32,39,38,36);(1,4,5,3)(2,8,6,7)(9,13)(10,17)(11,20)(12,15)(14,22)(16,23)(18,19)(21,24)(25,29,30,33)(26,34,31,28)(27,36,37,39)(32,40,38,35)Polycyclic presentation:
2^9|b^2=gi,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^c=ehi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7549798);