Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1277 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26,31,37)(27,38,40,33)(28,34,43,52)(29,45,32,35)(30,36,46,48)(39,53,54,47)(41,51,55,49)(42,56,44,50);(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,31,40)(26,33,37,38)(28,41,43,55)(29,30,32,46)(34,49,52,51)(35,36,45,48)(39,42,54,44)(47,56,53,50);(2,5)(6,8)(25,28,29,42,31,43,32,44)(26,34,45,56,37,52,35,50)(27,39,46,55,40,54,30,41)(33,47,36,51,38,53,48,49);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,29,31,32)(26,35,37,45)(27,30,40,46)(28,42,43,44)(33,36,38,48)(34,50,52,56)(39,41,54,55)(47,51,53,49);(1,3,4,7)(2,6,5,8)(11,19)(14,17)(18,24)(22,23)(25,30,29,27,31,46,32,40)(26,36,45,38,37,48,35,33)(28,41,42,39,43,55,44,54)(34,51,56,53,52,49,50,47)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=hi,d^2=f,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6738430);