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)(10,14)(11,17)(12,20)(22,24)(25,26,29,30)(27,33,35,31)(28,32,36,37)(34,39,40,38)(43,49)(46,50)(51,55)(53,56);(3,8)(5,7)(11,17)(12,20)(15,18)(16,21)(26,30)(28,36)(31,33)(34,40)(41,42,45,48)(43,50,49,46)(44,47,52,54)(51,56,55,53);(9,10,13,14)(11,18,17,15)(12,16,20,21)(19,24,23,22)(25,26)(27,31)(28,37)(29,30)(32,36)(33,35)(34,39)(38,40)(41,42)(43,46)(44,47)(45,48)(49,50)(51,53)(52,54)(55,56);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39)(41,43)(42,46)(44,51)(45,49)(47,53)(48,50)(52,55)(54,56);(1,2)(3,5)(4,6)(7,8)(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,39)(35,40)(41,44,45,52)(42,47,48,54)(43,51,49,55)(46,53,50,56)Polycyclic presentation:
2^9|a^2=ghi,b^2=fh,c^2=fg,d^2=f,e^2=gh,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, 6087117);