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 | |
4445 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,18)(12,13)(14,15)(16,21)(17,22)(19,24)(20,23)(26,32)(30,40)(33,36)(34,37);(1,2,5,6)(3,7,4,8)(25,26)(27,36)(28,33)(29,34)(30,35)(31,32)(37,39)(38,40);(1,3,5,4)(2,7,6,8)(10,15)(12,17)(16,18)(19,23)(25,27,31,28)(26,33,32,36)(29,38,39,35)(30,37,40,34);(1,3,5,4)(2,7,6,8)(9,11,14,21)(10,16,15,18)(12,19,17,23)(13,20,22,24)(25,28,31,27)(26,33,32,36)(29,38,39,35)(30,34,40,37);(9,12,14,17)(10,13,15,22)(11,19,21,23)(16,20,18,24)(25,29)(26,34)(27,35)(28,38)(30,36)(31,39)(32,37)(33,40);(1,4,5,3)(2,8,6,7)(9,13)(10,17)(11,20)(12,15)(14,22)(16,23)(18,19)(21,24)(25,30,31,40)(26,35,32,38)(27,37,28,34)(29,36,39,33)Polycyclic presentation:
2^9|b^2=gi,c^2=g,d^2=gh,e^2=h,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, 7549788);