Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438975 |
| 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 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1401 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)(17,18,22,28,24,29,23,27)(19,30,21,39,33,41,32,25)(20,26,36,44,38,45,37,43)(31,46,35,48,47,42,34,40);(2,5)(6,8)(17,19,24,33)(18,25,29,39)(20,34,38,35)(21,22,32,23)(26,42,45,46)(27,30,28,41)(31,36,47,37)(40,44,48,43);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20,22,36,24,38,23,37)(18,26,28,44,29,45,27,43)(19,31,21,35,33,47,32,34)(25,40,30,46,39,48,41,42);(17,21,23,19,24,32,22,33)(18,25,27,41,29,39,28,30)(20,35,37,31,38,34,36,47)(26,40,43,42,45,48,44,46);(9,11,12,15)(10,14,13,16)(17,22,24,23)(18,27,29,28)(19,32,33,21)(20,36,38,37)(25,30,39,41)(26,43,45,44)(31,34,47,35)(40,46,48,42)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=f,d^2=h,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372237);