Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1280 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18)(19,24)(20,25)(21,36)(22,27)(23,28)(26,37)(29,38)(30,39)(31,40)(32,41)(33,42)(34,44)(35,43)(45,47)(46,48);(9,11)(10,13)(12,16)(14,15)(17,19,23,30)(18,24,28,39)(20,31,33,34)(21,22,37,32)(25,40,42,44)(26,41,36,27)(29,45,46,35)(38,47,48,43);(2,5)(6,8)(10,14)(11,16)(17,20,32,45,23,33,22,35)(18,25,41,47,28,42,27,43)(19,29,37,34,30,46,21,31)(24,38,26,44,39,48,36,40);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(20,33)(25,42)(29,46)(31,34)(35,45)(38,48)(40,44)(43,47);(17,21,22,30,23,37,32,19)(18,26,27,24,28,36,41,39)(20,34,35,29,33,31,45,46)(25,40,43,48,42,44,47,38)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fhi,d^2=fg,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6736269);