Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| 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 | |
1317 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)(9,10,13,16)(11,17,18,14)(12,15,20,22)(19,23,24,21)(25,26,30,35,31,36,29,34)(27,32,40,48,38,46,39,47)(28,41,42,49,43,50,44,33)(37,51,53,55,52,56,54,45);(2,5)(6,8)(9,11)(10,14)(12,19)(13,18)(15,21)(16,17)(20,24)(22,23)(25,27,31,38)(26,32,36,46)(28,37,43,52)(29,39,30,40)(33,45,49,55)(34,47,35,48)(41,51,50,56)(42,53,44,54);(11,18)(14,17)(19,24)(21,23)(25,28,31,43)(26,33,36,49)(27,37,38,52)(29,42,30,44)(32,45,46,55)(34,41,35,50)(39,53,40,54)(47,51,48,56);(1,3,4,7)(2,6,5,8)(12,20)(15,22)(19,24)(21,23)(27,38)(32,46)(37,52)(39,40)(45,55)(47,48)(51,56)(53,54);(9,12,13,20)(10,15,16,22)(11,19,18,24)(14,21,17,23)(25,29,31,30)(26,34,36,35)(27,39,38,40)(28,42,43,44)(32,47,46,48)(33,41,49,50)(37,53,52,54)(45,51,55,56)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=i,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6451524);