Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470355 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1324 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)(3,7)(4,5)(6,8)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26,30,33)(27,35,38,51)(28,40,42,32)(29,44,36,34)(31,45,47,39)(37,53,41,50)(43,49,52,55)(46,54,48,56);(2,5)(6,7)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,27,36,45,30,38,29,39)(26,31,44,51,33,47,34,35)(28,41,52,54,42,37,43,56)(32,48,55,50,40,46,49,53);(1,3,4,8)(2,6,5,7)(11,17)(14,21)(18,23)(22,24)(25,28,30,42)(26,32,33,40)(27,37,38,41)(29,43,36,52)(31,46,47,48)(34,49,44,55)(35,50,51,53)(39,54,45,56);(11,17)(14,21)(18,23)(22,24)(27,38)(28,42)(31,47)(32,40)(35,51)(39,45)(43,52)(49,55);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(26,33)(31,47)(32,40)(34,44)(35,51)(46,48)(49,55)(50,53)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=g,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6405194);