Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535664 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#553 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1383 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,16)(13,15)(17,21)(18,20)(19,22)(23,24)(25,26)(27,36)(28,41)(29,34)(30,46)(31,35)(32,38)(33,44)(37,50)(39,49)(40,48)(42,47)(43,51)(45,56)(52,53)(54,55);(2,5)(6,8)(9,11,13,17)(10,14,15,21)(12,18,19,23)(16,20,22,24)(25,27,30,38,31,40,29,39)(26,32,34,36,35,49,46,48)(28,42,44,54,45,37,43,53)(33,50,41,55,51,47,56,52);(1,3,4,7)(2,6,5,8)(10,15)(11,17)(16,22)(18,23)(25,28,30,44,31,45,29,43)(26,33,34,41,35,51,46,56)(27,37,38,53,40,42,39,54)(32,47,36,52,49,50,48,55);(9,12)(10,16)(11,18)(13,19)(14,20)(15,22)(17,23)(21,24)(25,29,31,30)(26,34,35,46)(27,38,40,39)(28,43,45,44)(32,48,49,36)(33,41,51,56)(37,53,42,54)(47,55,50,52);(12,19)(16,22)(18,23)(20,24)(25,29,31,30)(26,34,35,46)(27,39,40,38)(28,43,45,44)(32,36,49,48)(33,41,51,56)(37,54,42,53)(47,52,50,55)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=f,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6475970);