Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2361 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
793 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,18,15,20,14,19)(11,21,23,16,25,32,24,33)(12,26,28,36,30,35,29,17)(22,31,27,34,39,40,38,37);(1,3,4,8)(2,5,6,7)(9,11,13,23,15,25,14,24)(10,16,19,21,20,33,18,32)(12,27,28,39,30,38,29,22)(17,34,35,31,36,37,26,40);(9,12,14,29,15,30,13,28)(10,17,18,26,20,36,19,35)(11,22,24,38,25,39,23,27)(16,31,32,34,33,40,21,37);(2,6)(5,7)(10,18,20,19)(16,32,33,21)(17,26,36,35)(31,34,40,37)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=f,d^2=fh,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 285581);