Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| 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 | |
1238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,7)(3,5,8,6)(12,19)(16,21)(17,23)(22,24)(25,26,30,27)(28,33,36,31)(29,32,37,34)(35,39,40,38)(41,42)(43,46)(44,49)(45,48)(47,52)(50,53)(51,55)(54,56);(1,3,4,8)(2,5,7,6)(10,14)(15,20)(16,21)(22,24)(26,27)(31,33)(32,34)(38,39)(42,43)(45,53)(47,49)(51,56);(1,3,4,8)(2,6,7,5)(9,10)(11,15)(12,16)(13,14)(17,22)(18,20)(19,21)(23,24)(25,27,30,26)(28,31,36,33)(29,32,37,34)(35,39,40,38)(41,43,46,42)(44,49,52,47)(45,48,53,50)(51,54,56,55);(1,3)(2,6)(4,8)(5,7)(9,11,13,18)(10,15,14,20)(12,17,19,23)(16,22,21,24)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39)(41,44,46,52)(42,47,43,49)(45,51,53,56)(48,54,50,55);(1,2)(3,6)(4,7)(5,8)(9,12)(10,16)(11,17)(13,19)(14,21)(15,22)(18,23)(20,24)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39)(41,45,46,53)(42,48,43,50)(44,51,52,56)(47,54,49,55)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,c^2=i,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6178598);