Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×#121 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1358 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)(3,5)(4,6)(7,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,31,36,30,35,44,51)(27,34,38,50,29,32,43,46)(28,39,41,53,40,33,47,48)(37,45,52,55,42,49,54,56);(1,3)(2,5)(4,7)(6,8)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,27,30,29)(26,32,35,34)(28,37,40,42)(31,38,44,43)(33,45,39,49)(36,46,51,50)(41,52,47,54)(48,55,53,56);(2,6)(3,7)(25,28,31,41,30,40,44,47)(26,33,36,48,35,39,51,53)(27,37,38,52,29,42,43,54)(32,45,46,55,34,49,50,56);(2,6)(5,8)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,29,30,27)(26,34,35,32)(28,37,40,42)(31,43,44,38)(33,45,39,49)(36,50,51,46)(41,52,47,54)(48,55,53,56);(10,16)(14,17)(15,24)(22,23)(27,29)(32,34)(37,42)(38,43)(45,49)(46,50)(52,54)(55,56)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=h,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cgi,d^b=dfi,e^b=ei,d^c=di,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6318953);