Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4462490 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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)(12,19)(16,21)(17,23)(22,24)(25,26)(27,30)(28,31)(29,34)(32,37)(33,36)(35,39)(38,40)(41,42,46,44)(43,47,49,51)(45,52,53,48)(50,56,55,54);(1,2,4,6)(3,5,7,8)(10,14)(15,20)(16,21)(22,24)(26,27)(31,33)(32,34)(38,39)(42,44)(43,49)(48,52)(50,55);(9,10)(11,15)(12,16)(13,14)(17,22)(18,20)(19,21)(23,24)(25,27,30,26)(28,31,36,33)(29,34,37,32)(35,38,40,39)(41,43)(42,47)(44,51)(45,50)(46,49)(48,54)(52,56)(53,55);(1,3)(2,5)(4,7)(6,8)(9,11,13,18)(10,15,14,20)(12,17,19,23)(16,22,21,24)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40)(41,44,46,42)(43,47,49,51)(45,52,53,48)(50,54,55,56);(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)(42,48,44,52)(43,50,49,55)(47,54,51,56)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=g,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfg,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4370491);