Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | #379 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1698 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,4)(5,6,7,8)(9,10)(11,21)(12,17)(13,20)(14,18)(15,19)(16,23)(22,36)(24,30)(25,31)(26,35)(27,34)(28,33)(29,39)(32,37)(38,40);(9,11,14,24,15,25,13,23)(10,16,20,31,19,30,18,21)(12,22,26,37,27,38,28,39)(17,32,33,36,34,29,35,40);(1,2,3,4)(9,12)(10,17)(11,22)(13,28)(14,26)(15,27)(16,29)(18,35)(19,34)(20,33)(21,36)(23,39)(24,37)(25,38)(30,32)(31,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,h^a=hi,c^b=cf,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31063);