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 | #2847 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4354 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)(12,17)(13,15)(14,24)(18,20)(19,23)(21,22)(26,31)(29,33)(32,38)(35,39);(1,2)(3,5)(4,6)(7,8)(9,11,15,19)(10,16,13,23)(12,20,21,14)(17,18,22,24)(25,26)(27,32)(28,29)(30,31)(33,37)(34,35)(36,38)(39,40);(10,13)(11,19)(12,21)(18,24)(26,31)(28,37)(32,38)(34,40);(1,3,4,7)(2,5,6,8)(9,12)(10,17)(11,20)(13,22)(14,19)(15,21)(16,18)(23,24)(25,27,30,36)(26,32,31,38)(28,34,37,40)(29,35,33,39);(9,13,15,10)(11,16,19,23)(12,22,21,17)(14,24,20,18)(25,28)(26,33)(27,34)(29,31)(30,37)(32,39)(35,38)(36,40);(9,14,15,20)(10,18,13,24)(11,12,19,21)(16,22,23,17)(25,29,30,33)(26,28,31,37)(27,35,36,39)(32,34,38,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,d^2=g,e^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7627699);