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 |
| prime radical | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4160 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(25,26,30,33)(27,34,28,31)(29,32,37,40)(35,39,36,38);(1,3,5,8)(2,6,4,7)(9,10,14,15)(11,13,17,22)(12,16,21,23)(18,20,19,24);(1,2,5,4)(3,6,8,7)(10,15)(13,22)(16,23)(20,24)(25,27,30,28)(26,31,33,34)(29,36,37,35)(32,39,40,38);(9,11,14,17)(10,13,15,22)(12,19,21,18)(16,24,23,20)(25,26)(27,31)(28,34)(29,32)(30,33)(35,38)(36,39)(37,40);(9,12,14,21)(10,16,15,23)(11,18,17,19)(13,20,22,24)(25,28,30,27)(26,31,33,34)(29,36,37,35)(32,38,40,39);(1,4,5,2)(3,7,8,6)(9,13)(10,17)(11,15)(12,20)(14,22)(16,19)(18,23)(21,24)(25,29,30,37)(26,32,33,40)(27,35,28,36)(31,38,34,39)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=gh,d^2=i,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7559987);