Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(3,7)(5,8)(10,14)(11,17)(16,21)(18,23)(25,26,29,30)(27,33,34,31)(28,32,37,36)(35,39,40,38)(43,49)(44,51)(46,52)(47,54);(11,17)(15,20)(18,23)(22,24)(26,30)(27,34)(32,36)(35,40)(41,42,45,48)(43,46,49,52)(44,47,51,54)(50,53,55,56);(1,2,4,6)(3,5,7,8)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,26)(27,31)(28,36)(29,30)(32,37)(33,34)(35,39)(38,40)(41,42)(43,46)(44,47)(45,48)(49,52)(50,53)(51,54)(55,56);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,27,29,34)(26,31,30,33)(28,35,37,40)(32,38,36,39)(41,43,45,49)(42,46,48,52)(44,50,51,55)(47,53,54,56);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,28,29,37)(26,32,30,36)(27,35,34,40)(31,38,33,39)(41,44)(42,47)(43,50)(45,51)(46,53)(48,54)(49,55)(52,56)Polycyclic presentation:
2^9|a^2=f,b^2=fh,c^2=hi,d^2=h,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4546079);