Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle, prime radical | C23 | C2×C4×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
626 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,7)(5,8)(9,10,13,16)(11,17,20,24)(12,21,18,15)(14,22,23,19)(25,26,30,33,31,34,29,38)(27,32,28,37,35,39,36,40);(1,3,6,2)(4,7,5,8)(25,27,29,36,31,35,30,28)(26,32,38,40,34,39,33,37);(1,4,6,5)(2,7,3,8)(9,11,18,22,13,20,12,19)(10,14,21,24,16,23,15,17)(25,28,30,35,31,36,29,27)(26,32,38,40,34,39,33,37);(1,5,6,4)(2,7,3,8)(27,35)(28,36)(32,39)(37,40)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=efh,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=di,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 126652);