Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499792 |
| 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 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1388 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)(25,26,30,34)(27,35,37,50)(28,32,40,48)(29,42,44,33)(31,43,46,38)(36,51,39,53)(41,56,55,49)(45,54,47,52);(1,3,4,7)(2,5,6,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,27)(26,31)(28,39)(29,43)(30,37)(32,47)(33,50)(34,46)(35,42)(36,40)(38,44)(41,52)(45,48)(49,51)(53,56)(54,55);(2,6)(5,8)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,28,30,40)(26,32,34,48)(27,36,37,39)(29,41,44,55)(31,45,46,47)(33,49,42,56)(35,51,50,53)(38,52,43,54);(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(28,40)(32,48)(36,39)(41,55)(45,47)(49,56)(51,53)(52,54);(10,16)(14,17)(15,24)(22,23)(27,37)(28,40)(31,46)(32,48)(35,50)(38,43)(41,55)(49,56)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=h,d^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6439220);