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:
(1,2,4,6)(3,7,8,5)(11,17)(15,20)(18,23)(22,24)(26,30)(27,33)(32,37)(35,39)(42,46)(43,49)(48,51)(50,55);(3,8)(5,7)(10,14)(11,17)(16,21)(18,23)(26,30)(27,33)(28,36)(38,40)(41,42,45,46)(43,47,49,53)(44,51,52,48)(50,56,55,54);(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,26,29,30)(27,34,33,31)(28,32,36,37)(35,40,39,38)(41,42)(43,47)(44,51)(45,46)(48,52)(49,53)(50,56)(54,55);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,27,29,33)(26,31,30,34)(28,35,36,39)(32,38,37,40)(41,43,45,49)(42,47,46,53)(44,50,52,55)(48,54,51,56);(1,2)(3,5)(4,6)(7,8)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,28)(26,32)(27,35)(29,36)(30,37)(31,38)(33,39)(34,40)(41,44,45,52)(42,48,46,51)(43,50,49,55)(47,54,53,56)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=fg,d^2=g,e^2=fgh,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, 6085589);