Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23876 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#385 |
| prime radical | C24 | C22×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
987 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)(3,7)(4,5)(6,8)(9,10,14,18,15,20,13,19)(11,16,23,31,24,32,22,36)(12,25,27,17,29,34,28,35)(21,37,26,30,38,33,39,40);(2,5)(6,7)(9,11,14,23,15,24,13,22)(10,16,18,31,20,32,19,36)(12,26,27,38,29,39,28,21)(17,33,34,40,35,37,25,30);(9,12,13,28,15,29,14,27)(10,17,19,25,20,35,18,34)(11,21,23,26,24,38,22,39)(16,30,31,33,32,40,36,37);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^b=gi,f^c=fi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275485);