Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C22×#107 |
| prime radical | C24 | #107 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
494 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)(11,20)(12,17)(13,27)(14,29)(15,19)(16,24)(18,21)(22,33)(23,34)(25,39)(26,32)(28,40)(30,37)(31,38)(35,36);(9,11,21,36,14,25,28,37,15,22,26,38,12,23,13,24)(10,16,27,34,17,31,32,33,19,30,40,39,29,35,18,20);(2,5)(6,8)(11,22)(16,30)(20,33)(23,25)(24,37)(31,35)(34,39)(36,38);(9,12,15,14)(10,17,19,29)(11,23,22,25)(13,26,28,21)(16,31,30,35)(18,27,32,40)(20,34,33,39)(24,38,37,36)Polycyclic presentation:
2^9|a^2=f,b^2=eg,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=ch,e^a=eg,g^a=gi,c^b=ci104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390601);