Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469983 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1445 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)(3,5)(4,6)(7,8)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26,30,33)(27,35,37,50)(28,32,40,48)(29,42,44,34)(31,43,46,38)(36,51,39,53)(41,56,55,49)(45,54,47,52);(1,3,4,7)(2,5,6,8)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,27)(26,31)(28,39)(29,43)(30,37)(32,47)(33,46)(34,50)(35,42)(36,40)(38,44)(41,52)(45,48)(49,51)(53,56)(54,55);(2,6)(5,8)(11,17)(14,21)(18,23)(22,24)(25,28,30,40)(26,32,33,48)(27,36,37,39)(29,41,44,55)(31,45,46,47)(34,49,42,56)(35,51,50,53)(38,52,43,54);(11,17)(14,21)(18,23)(22,24)(27,37)(28,40)(31,46)(32,48)(35,50)(38,43)(41,55)(49,56);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(26,33)(28,40)(31,46)(34,42)(35,50)(36,39)(41,55)(52,54)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=i,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=di,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6404497);