Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C22×C4⋊C8 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
587 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,19,14,18,27,34,15,17,28,33,12,24,26,35)(11,20,22,36,21,16,29,31,23,32,37,40,25,30,38,39);(9,11,13,22,14,21,27,29,15,23,28,37,12,25,26,38)(10,16,19,31,18,32,34,40,17,30,33,39,24,20,35,36);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(10,17)(16,30)(18,24)(19,33)(20,32)(31,39)(34,35)(36,40)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=f,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,e^b=ei128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402651);