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)(17,18,23,29,24,30,22,28)(19,31,21,27,33,45,38,25)(20,26,36,43,37,44,35,42)(32,40,34,41,46,48,47,39);(2,5)(6,8)(17,19,24,33)(18,25,30,27)(20,34,37,47)(21,23,38,22)(26,40,44,48)(28,31,29,45)(32,35,46,36)(39,42,41,43);(10,13)(14,16)(17,20,23,36,24,37,22,35)(18,26,29,43,30,44,28,42)(19,32,21,34,33,46,38,47)(25,39,31,40,27,41,45,48);(1,3,4,7)(2,6,5,8)(17,21,23,33,24,38,22,19)(18,27,29,45,30,25,28,31)(20,34,36,46,37,47,35,32)(26,41,43,48,44,39,42,40);(9,11,12,15)(10,14,13,16)(17,22,24,23)(18,28,30,29)(19,21,33,38)(20,35,37,36)(25,31,27,45)(26,42,44,43)(32,34,46,47)(39,40,41,48)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,d^2=f,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^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6369889);