Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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 | |
1240 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)(3,5,7,8)(10,14)(11,17)(12,20)(22,24)(26,30)(27,33)(32,37)(35,39)(42,46)(44,52)(47,49)(50,56);(10,14)(12,20)(15,18)(19,23)(28,36)(32,37)(35,39)(38,40)(41,42,45,46)(43,49,51,47)(44,53,52,48)(50,54,56,55);(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,26,29,30)(27,34,33,31)(28,32,36,37)(35,40,39,38)(41,42)(43,47)(44,53)(45,46)(48,52)(49,51)(50,55)(54,56);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(25,27,29,33)(26,31,30,34)(28,35,36,39)(32,38,37,40)(41,43)(42,47)(44,50)(45,51)(46,49)(48,54)(52,56)(53,55);(1,3,4,7)(2,5,6,8)(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,28)(26,32)(27,35)(29,36)(30,37)(31,38)(33,39)(34,40)(41,44)(42,48)(43,50)(45,52)(46,53)(47,54)(49,55)(51,56)Polycyclic presentation:
2^9|a^2=fgi,b^2=f,c^2=gh,d^2=gh,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5093486);