Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469956 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,8)(4,6)(5,7)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26,30,33)(27,35,37,50)(28,39,41,32)(29,43,45,34)(31,44,47,38)(36,52,40,51)(42,49,56,54)(46,53,48,55);(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,27,30,37)(26,31,33,47)(28,40,41,36)(29,44,45,38)(32,48,39,46)(34,50,43,35)(42,53,56,55)(49,51,54,52);(1,3,5,4)(2,6,7,8)(11,17)(14,21)(18,23)(22,24)(25,28,30,41)(26,32,33,39)(27,36,37,40)(29,42,45,56)(31,46,47,48)(34,49,43,54)(35,51,50,52)(38,53,44,55);(1,4,5,3)(2,6,7,8)(11,17)(14,21)(18,23)(22,24)(27,37)(28,41)(31,47)(32,39)(35,50)(38,44)(42,56)(49,54);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(26,33)(31,47)(32,39)(34,43)(35,50)(46,48)(49,54)(51,52)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=g,d^2=gi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6405322);