Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #117 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C23 | #432 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
678 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)(6,8)(9,10,14,21,15,20,31,36,16,22,30,18,13,19,12,29)(11,23,26,38,27,17,24,33,28,34,32,40,25,35,39,37);(2,4)(9,11)(10,17,22,35)(12,24)(13,25)(14,32)(15,27)(16,28)(18,33,21,37)(19,23,20,34)(26,30)(29,40,36,38)(31,39);(5,6,7,8)(9,12,13,30,16,31,15,14)(10,18,20,29,22,21,19,36)(11,24,25,26,28,39,27,32)(17,33,34,40,35,37,23,38)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=e,d^2=i,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,g^a=gi,f^b=fi128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54150);