Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ 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:
(1,2)(3,4)(5,6)(7,13)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22,27,33)(23,34,36,29)(24,37,39,30)(25,40,28,31)(26,32,43,49)(35,50,51,45)(38,48,42,46)(41,52,44,47);(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,23,27,36)(22,29,33,34)(24,28,39,25)(26,42,43,38)(30,31,37,40)(32,48,49,46)(35,44,51,41)(45,47,50,52);(2,4)(21,24,28,36,27,39,25,23)(22,30,31,34,33,37,40,29)(26,38,44,51,43,42,41,35)(32,46,47,50,49,48,52,45);(6,11)(10,13)(12,19)(18,20)(21,25,27,28)(22,31,33,40)(23,24,36,39)(26,41,43,44)(29,37,34,30)(32,47,49,52)(35,38,51,42)(45,48,50,46);(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(21,26,28,44,27,43,25,41)(22,32,31,47,33,49,40,52)(23,35,24,38,36,51,39,42)(29,45,30,46,34,50,37,48)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=h,d^2=i,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6716715);