Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1280 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)(3,6)(4,8)(5,7)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26,31,36)(27,32,40,48)(28,33,30,35)(29,34,43,51)(37,45,39,47)(38,46,53,55)(41,49,44,52)(42,50,54,56);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,31,40)(26,32,36,48)(28,39,30,37)(29,42,43,54)(33,47,35,45)(34,50,51,56)(38,44,53,41)(46,52,55,49);(1,4,5,3)(2,6,7,8)(27,37)(28,30)(32,45)(33,35)(38,42)(39,40)(41,44)(46,50)(47,48)(49,52)(53,54)(55,56);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,28,31,30)(26,33,36,35)(27,37,40,39)(29,41,43,44)(32,45,48,47)(34,49,51,52)(38,42,53,54)(46,50,55,56);(11,19)(14,17)(18,24)(22,23)(25,29,31,43)(26,34,36,51)(27,38,40,53)(28,41,30,44)(32,46,48,55)(33,49,35,52)(37,42,39,54)(45,50,47,56)Polycyclic presentation:
2^9|a^2=fi,b^2=gi,c^2=g,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6598219);