Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1737 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,18,23,29)(19,25,33,43)(20,34,37,26)(21,27,24,30)(22,40,31,28)(32,42,41,48)(35,47,39,44)(36,46,38,45);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,33)(18,25,29,43)(20,35,37,39)(21,32,24,41)(22,38,31,36)(26,44,34,47)(27,42,30,48)(28,46,40,45);(10,14)(13,15)(17,20,23,37)(18,26,29,34)(19,31,33,22)(21,38,24,36)(25,40,43,28)(27,46,30,45)(32,39,41,35)(42,47,48,44);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,23,24)(18,27,29,30)(19,32,33,41)(20,36,37,38)(22,39,31,35)(25,42,43,48)(26,45,34,46)(28,47,40,44);(2,5)(6,8)(11,16)(13,15)(17,22,23,31)(18,28,29,40)(19,20,33,37)(21,39,24,35)(25,26,43,34)(27,47,30,44)(32,36,41,38)(42,45,48,46)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,d^2=fg,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6606525);