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 | #2397 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1489 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)(3,7,8,6)(9,10)(11,15)(12,13)(14,16)(17,18,23,27)(19,28,31,24)(20,32,37,48)(21,38,30,41)(22,39,34,26)(25,42,45,36)(29,46,47,43)(33,44,35,40);(2,5)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,23,31)(18,24,27,28)(20,33,37,35)(21,22,30,34)(25,43,45,46)(26,41,39,38)(29,42,47,36)(32,40,48,44);(10,13)(14,15)(17,20,34,42,23,37,22,36)(18,25,39,48,27,45,26,32)(19,29,30,35,31,47,21,33)(24,40,41,46,28,44,38,43);(9,11,12,16)(10,14,13,15)(17,21,22,31,23,30,34,19)(18,24,26,38,27,28,39,41)(20,35,36,29,37,33,42,47)(25,44,32,46,45,40,48,43);(1,3,4,8)(2,6,5,7)(17,22,23,34)(18,26,27,39)(19,30,31,21)(20,36,37,42)(24,41,28,38)(25,32,45,48)(29,35,47,33)(40,46,44,43)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=hi,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, 7503847);