Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1268 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,25)(20,34)(21,27)(22,28)(23,29)(24,30)(26,31)(32,40)(33,41)(35,47)(36,43)(37,42)(38,44)(39,45)(46,48);(9,11)(10,13)(12,16)(14,15)(17,19,23,33)(18,25,29,41)(20,21,31,24)(22,37,38,35)(26,30,34,27)(28,42,44,47)(32,36,46,39)(40,43,48,45);(2,5)(6,8)(11,16)(13,15)(17,20,24,19,23,31,21,33)(18,26,30,41,29,34,27,25)(22,35,39,32,38,37,36,46)(28,42,45,48,44,47,43,40);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,23,24)(18,27,29,30)(19,31,33,20)(22,36,38,39)(25,26,41,34)(28,43,44,45)(32,37,46,35)(40,42,48,47);(17,22,21,36,23,38,24,39)(18,28,27,43,29,44,30,45)(19,32,20,35,33,46,31,37)(25,40,34,47,41,48,26,42)Polycyclic presentation:
2^9|b^2=i,c^2=fh,d^2=g,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6733646);