Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24283 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C4×#121 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
786 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(9,10,14,19,15,20,13,16)(11,21,24,33,25,30,23,17)(12,26,29,34,27,31,28,18)(22,36,39,40,37,32,38,35);(1,2,4,6)(3,7,8,5)(10,16,20,19)(12,27)(17,30,33,21)(18,26,34,31)(22,37)(28,29)(32,35,36,40)(38,39);(1,3,4,8)(2,5,6,7)(9,11)(10,17)(12,22)(13,23)(14,24)(15,25)(16,30)(18,32)(19,21)(20,33)(26,35)(27,37)(28,38)(29,39)(31,40)(34,36);(9,12,13,28,15,27,14,29)(10,18,19,26,20,34,16,31)(11,22,23,38,25,37,24,39)(17,32,21,35,33,36,30,40)Polycyclic presentation:
2^9|a^2=h,b^2=egh,c^2=g,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^b=di,f^c=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 277268);