Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32935 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#3054 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1892 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,4)(5,6)(7,8)(9,10,14,18,15,19,13,29)(11,20,24,31,22,34,23,16)(12,25,28,32,26,39,27,17)(21,35,37,40,38,30,36,33);(1,2,3,4)(9,11)(10,16,19,31)(12,21)(13,23)(14,24)(15,22)(17,30,32,35)(18,20,29,34)(25,33,39,40)(26,38)(27,36)(28,37);(9,12,13,27,15,26,14,28)(10,17,18,25,19,32,29,39)(11,21,23,36,22,38,24,37)(16,30,20,33,31,35,34,40);(6,8)(11,22)(12,26)(16,31)(17,32)(20,34)(23,24)(25,39)(27,28)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,d^b=di,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300207);