Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438991 |
| 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×#466 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1417 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,27,24,29,23,28)(19,30,20,25,33,45,32,40)(21,26,36,42,38,44,37,43)(31,41,34,39,46,48,35,47);(2,5)(6,8)(17,19,24,33)(18,25,29,40)(20,22,32,23)(21,35,38,34)(26,41,44,48)(27,30,28,45)(31,36,46,37)(39,42,47,43);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20,23,19,24,32,22,33)(18,25,28,30,29,40,27,45)(21,34,37,31,38,35,36,46)(26,39,43,41,44,47,42,48);(17,21,22,36,24,38,23,37)(18,26,27,42,29,44,28,43)(19,31,20,34,33,46,32,35)(25,39,45,48,40,47,30,41);(9,11,12,15)(10,14,13,16)(17,22,24,23)(18,27,29,28)(19,32,33,20)(21,36,38,37)(25,30,40,45)(26,42,44,43)(31,35,46,34)(39,41,47,48)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=f,d^2=hi,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372047);