Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884969 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2414 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1408 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,5)(3,7)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26,30,34)(27,35,36,50)(28,32,40,48)(29,42,44,33)(31,43,45,38)(37,51,39,53)(41,56,55,49)(46,54,47,52);(1,2,4,5)(3,6,7,8)(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(27,36)(31,45)(35,50)(37,39)(38,43)(46,47)(51,53)(52,54);(2,5)(6,8)(12,21)(15,20)(18,24)(22,23)(25,27,30,36)(26,31,34,45)(28,39,40,37)(29,43,44,38)(32,47,48,46)(33,50,42,35)(41,52,55,54)(49,51,56,53);(1,3,4,7)(2,6,5,8)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,28,30,40)(26,32,34,48)(27,37,36,39)(29,41,44,55)(31,46,45,47)(33,49,42,56)(35,51,50,53)(38,52,43,54);(11,19)(14,17)(18,24)(22,23)(28,40)(32,48)(37,39)(41,55)(46,47)(49,56)(51,53)(52,54)Polycyclic presentation:
2^9|a^2=fh,b^2=f,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,d^c=dfh,g^c=gi,e^d=ei59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6824331);