Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6358426 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1578 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,9)(3,7,5,13)(4,11,12,15)(8,14,16,10)(17,18,23,28)(19,29,32,43)(20,25,35,42)(21,27,22,26)(24,36,39,31)(30,45,47,40)(33,48,46,38)(34,44,37,41);(1,3,6,5)(2,7,9,13)(4,10,12,14)(8,15,16,11)(17,19,23,32)(18,24,28,39)(20,33,35,46)(21,31,22,36)(25,40,42,45)(26,29,27,43)(30,37,47,34)(38,44,48,41);(1,4,6,12)(2,8,9,16)(3,10,5,14)(7,15,13,11);(1,3,6,5)(2,7,9,13)(4,10,12,14)(8,15,16,11)(17,20)(18,25)(19,30)(21,34)(22,37)(23,35)(24,38)(26,41)(27,44)(28,42)(29,45)(31,46)(32,47)(33,36)(39,48)(40,43);(17,21,23,22)(18,26,28,27)(19,31,32,36)(20,34,35,37)(24,29,39,43)(25,41,42,44)(30,46,47,33)(38,45,48,40)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=i,d^2=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,d^b=df,f^b=fh,g^c=gi,f^d=fh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6265741);