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 | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4488 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,16)(12,20)(13,17)(14,19)(15,21)(18,22)(23,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,37)(36,40)(38,39);(1,2)(3,5)(4,6)(7,8)(26,31)(27,33)(34,39)(35,40);(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,30,33)(26,32,31,28)(29,35,38,40)(34,37,39,36);(9,12,13,21)(10,15,17,20)(11,18,19,24)(14,22,16,23);(1,3,4,7)(2,5,6,8)(9,11)(10,16)(12,18)(13,19)(14,17)(15,23)(20,22)(21,24)(25,28,30,32)(26,33,31,27)(29,37,38,36)(34,35,39,40);(9,10)(11,14)(12,15)(13,17)(16,19)(18,22)(20,21)(23,24)(25,29)(26,34)(27,35)(28,36)(30,38)(31,39)(32,37)(33,40)Polycyclic presentation:
2^9|a^2=gi,c^2=i,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7562254);