Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432843 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1319 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26,30,33)(27,31,36,46)(28,38,41,50)(29,42,44,34)(32,43,49,40)(35,51,39,56)(37,54,53,47)(45,55,48,52);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,27,30,36)(26,31,33,46)(28,39,41,35)(29,37,44,53)(32,48,49,45)(34,47,42,54)(38,56,50,51)(40,55,43,52);(11,17)(14,21)(18,23)(22,24)(25,28,30,41)(26,32,33,49)(27,35,36,39)(29,43,44,40)(31,45,46,48)(34,50,42,38)(37,55,53,52)(47,56,54,51);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(26,33)(31,46)(32,49)(34,42)(38,50)(45,48)(47,54)(51,56);(1,3,4,7)(2,6,5,8)(27,36)(31,46)(35,39)(37,53)(45,48)(47,54)(51,56)(52,55)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=i,d^2=hi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,e^b=ei,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6343459);