Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1375 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,7,4,8)(9,10)(11,15)(12,13)(14,16)(17,18,23,29)(19,27,33,25)(20,34,37,44)(21,39,32,42)(22,30,24,28)(26,36,45,38)(31,47,48,43)(35,46,40,41);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,33)(18,25,29,27)(20,35,37,40)(21,22,32,24)(26,43,45,47)(28,42,30,39)(31,36,48,38)(34,41,44,46);(1,3,5,4)(2,7,6,8)(10,13)(14,15)(17,20,22,36,23,37,24,38)(18,26,28,44,29,45,30,34)(19,31,21,40,33,48,32,35)(25,41,39,47,27,46,42,43);(9,11,12,16)(10,14,13,15)(17,21,24,19,23,32,22,33)(18,27,30,39,29,25,28,42)(20,35,38,48,37,40,36,31)(26,41,34,43,45,46,44,47);(1,4,5,3)(2,7,6,8)(17,22,23,24)(18,28,29,30)(19,32,33,21)(20,36,37,38)(25,42,27,39)(26,44,45,34)(31,35,48,40)(41,43,46,47)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fh,d^2=g,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7504899);