Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9091 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1436 |
| prime radical | C24 | C4×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
542 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(9,10,14,18,15,19,13,28)(11,20,23,32,24,31,22,16)(12,25,26,17,27,33,29,34)(21,36,37,40,39,30,38,35);(1,2,3,6,4,5,7,8)(9,11)(10,16,19,32)(12,21)(13,22)(14,23)(15,24)(17,30,34,36)(18,20,28,31)(25,40,33,35)(26,37)(27,39)(29,38);(2,5)(6,8)(9,12,14,26,15,27,13,29)(10,17,18,33,19,34,28,25)(11,21,23,37,24,39,22,38)(16,30,20,35,32,36,31,40)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,c^b=ci,e^b=eh,d^c=dh,e^c=eh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 16733);