Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #42806 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4116 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 56 points:
(2,5)(3,4)(10,14)(12,19)(15,20)(18,23)(25,26,31,35)(27,36,33,29)(28,32,39,46)(30,44,45,34)(37,47,40,41)(38,43,49,50)(42,55,52,48)(51,54,53,56);(1,2,6,5)(3,8,4,7)(11,17)(15,20)(18,23)(22,24)(27,33)(28,39)(29,36)(32,46)(38,49)(42,52)(43,50)(48,55);(1,3,6,4)(2,7,5,8)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,27,31,33)(26,29,35,36)(28,40,39,37)(30,38,45,49)(32,47,46,41)(34,43,44,50)(42,53,52,51)(48,56,55,54);(1,4,6,3)(2,8,5,7)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,28,31,39)(26,32,35,46)(27,37,33,40)(29,41,36,47)(30,42,45,52)(34,48,44,55)(38,51,49,53)(43,54,50,56);(1,5,6,2)(3,8,4,7)(9,12)(10,16)(11,18)(13,19)(14,21)(15,22)(17,23)(20,24)(25,29)(26,33)(27,35)(28,41)(30,43)(31,36)(32,40)(34,49)(37,46)(38,44)(39,47)(42,54)(45,50)(48,53)(51,55)(52,56);(25,30,31,45)(26,34,35,44)(27,38,33,49)(28,42,39,52)(29,43,36,50)(32,48,46,55)(37,51,40,53)(41,54,47,56)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=h,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8022616);