Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 | C2×#540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1709 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)(5,6,9,12)(7,13,15,10)(8,16,17,11)(14,18,20,19)(21,22,26,30)(23,27,33,39)(24,28,35,41)(25,29,36,42)(31,37,44,48)(32,38,45,49)(34,40,46,50)(43,47,51,52);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(23,31)(24,34)(25,36)(27,37)(28,40)(29,42)(32,45)(33,44)(35,46)(38,49)(39,48)(41,50);(2,4)(21,23,25,31,26,33,36,44)(22,27,29,37,30,39,42,48)(24,32,34,43,35,45,46,51)(28,38,40,47,41,49,50,52);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(24,35)(28,41)(32,45)(34,46)(38,49)(40,50)(43,51)(47,52);(7,15)(10,13)(14,20)(18,19)(21,24,36,46,26,35,25,34)(22,28,42,50,30,41,29,40)(23,32,44,51,33,45,31,43)(27,38,48,52,39,49,37,47)Polycyclic presentation:
2^9|a^2=fi,c^2=h,d^2=f,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6715996);