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 | #37553 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4292 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)(9,10,14,17)(11,12,19,15)(13,21,22,16)(18,23,24,20)(29,37)(32,39)(35,38)(36,40);(1,3,5,8)(2,6,4,7)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,38)(37,39);(9,11)(10,15)(12,17)(13,18)(14,19)(16,23)(20,21)(22,24)(25,27,30,31)(26,28,33,34)(29,35,37,38)(32,36,39,40);(9,12,14,15)(10,11,17,19)(13,20,22,23)(16,18,21,24)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,39)(38,40);(1,4,5,2)(3,7,8,6)(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,23)(22,24)(25,28)(26,31)(27,33)(29,36)(30,34)(32,38)(35,39)(37,40);(9,13,14,22)(10,16,17,21)(11,18,19,24)(12,20,15,23)(25,29,30,37)(26,32,33,39)(27,35,31,38)(28,36,34,40)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=i,d^2=h,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7555649);