Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1646 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6,9,12)(7,13,15,10)(8,16,17,11)(14,18,20,19)(21,22,28,34)(23,29,37,45)(24,30,35,43)(25,31,27,33)(26,32,42,50)(36,44,51,52)(38,46,40,48)(39,47,41,49);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,23,28,37)(22,29,34,45)(24,25,35,27)(26,40,42,38)(30,31,43,33)(32,48,50,46)(36,39,51,41)(44,47,52,49);(1,2,3,4)(21,24,27,23,28,35,25,37)(22,30,33,29,34,43,31,45)(26,40,41,51,42,38,39,36)(32,48,49,52,50,46,47,44);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(21,25,28,27)(22,31,34,33)(23,35,37,24)(26,39,42,41)(29,43,45,30)(32,47,50,49)(36,40,51,38)(44,48,52,46);(7,15)(10,13)(14,20)(18,19)(21,26,27,41,28,42,25,39)(22,32,33,49,34,50,31,47)(23,36,35,40,37,51,24,38)(29,44,43,48,45,52,30,46)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=gh,d^2=fi,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6722583);