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)(13,19)(14,18)(15,20)(16,23)(22,33)(24,32)(25,31)(26,39)(27,36)(28,34)(29,35)(30,37)(38,40);(9,11,14,24,15,25,13,23)(10,16,19,31,20,32,18,21)(12,26,27,22,29,37,28,38)(17,33,34,39,35,40,36,30);(1,2,3,4)(9,12)(10,17)(11,22)(13,27)(14,28)(15,29)(16,30)(18,34)(19,36)(20,35)(21,33)(23,37)(24,26)(25,38)(31,40)(32,39)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=i,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,h^a=hi,c^b=cf,f^c=fi,h^c=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31148);