Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| 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 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1308 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,26,29,34,30,33,42,47)(27,31,37,45,36,44,52,55)(28,38,39,46,40,48,41,32)(35,49,51,54,50,56,53,43);(2,6)(7,8)(10,14)(11,17)(16,21)(18,23)(25,27,30,36)(26,31,33,44)(28,35,40,50)(29,37,42,52)(32,43,46,54)(34,45,47,55)(38,49,48,56)(39,51,41,53);(1,3,5,4)(2,7,6,8)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,28,30,40)(26,32,33,46)(27,35,36,50)(29,41,42,39)(31,43,44,54)(34,48,47,38)(37,53,52,51)(45,56,55,49);(1,4,5,3)(2,8,6,7)(12,19)(16,21)(18,23)(22,24)(27,36)(31,44)(35,50)(37,52)(43,54)(45,55)(49,56)(51,53);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(26,33)(31,44)(32,46)(34,47)(38,48)(43,54)(45,55)(49,56)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=fi,d^2=fh,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,g^c=gi,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6457817);