Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #118289 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1439 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
901 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,20)(12,24,27,17)(13,28)(14,18)(15,19)(16,22)(21,37,39,34)(23,33)(25,31,38,40)(26,35,30,36)(29,32);(1,2)(3,7,8,5)(4,6)(9,11,13,22,15,23,14,29)(10,16,18,20,19,32,28,33)(12,25,26,21,27,38,30,39)(17,34,35,40,24,37,36,31);(1,3)(2,5)(4,8)(6,7)(9,12)(10,17)(11,21)(13,26)(14,30)(15,27)(16,31)(18,35)(19,24)(20,34)(22,38)(23,39)(25,29)(28,36)(32,40)(33,37)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,f^b=fhi,d^c=di,f^c=fh,g^c=gh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 25485);