Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | #386 |
| prime radical | C25 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,4)(9,10)(11,20)(12,17)(13,26)(14,18)(15,19)(16,22)(21,36)(23,31)(24,33)(25,34)(27,30)(28,32)(29,37)(35,38)(39,40);(5,6,7,8)(9,11)(10,16)(12,21)(13,27)(14,22)(15,23)(17,29)(18,31)(19,30)(20,26)(24,39)(25,38)(28,37)(32,35)(33,36)(34,40);(9,12,14,24,15,25,13,28)(10,17,18,33,19,34,26,32)(11,21,22,39,23,38,27,37)(16,29,31,36,30,40,20,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,d^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,d^b=di,e^b=ei,d^c=di128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30685);