Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519083 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1825 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,6)(3,7,4,8)(9,10,15,20)(11,22,24,17)(12,25,27,36)(13,28,30,34)(14,21,16,19)(18,29,35,31)(23,37,38,39)(26,33,40,32);(2,6)(7,8)(9,11,15,24)(10,17,20,22)(12,16,27,14)(13,23,30,38)(18,33,35,32)(19,25,21,36)(26,31,40,29)(28,39,34,37);(1,3,5,4)(2,7,6,8)(9,12,16,24,15,27,14,11)(10,17,21,25,20,22,19,36)(13,26,31,38,30,40,29,23)(18,32,28,39,35,33,34,37);(9,13,14,29,15,30,16,31)(10,18,19,34,20,35,21,28)(11,23,27,40,24,38,12,26)(17,32,36,37,22,33,25,39);(1,4,5,3)(2,7,6,8)(9,14,15,16)(10,19,20,21)(11,12,24,27)(13,29,30,31)(17,25,22,36)(18,34,35,28)(23,26,38,40)(32,39,33,37)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fgi,d^2=gi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7472020);