Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C22×#107 |
| prime radical | C24 | C2×Q16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
553 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)(9,10,13,19,15,17,14,29)(11,20,22,37,23,36,30,35)(12,24,26,34,28,39,27,18)(16,25,32,38,31,40,33,21);(9,11,13,22,15,23,14,30)(10,16,19,32,17,31,29,33)(12,25,26,38,28,40,27,21)(18,35,24,20,34,37,39,36);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(10,17)(16,31)(18,34)(19,29)(20,36)(24,39)(32,33)(35,37)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=f,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,h^a=hi,e^b=eh,h^b=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402667);