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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1625 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)(5,6,10,13)(7,14,16,11)(8,17,9,12)(15,19,18,20)(21,22,27,26)(23,28,33,41)(24,29,37,36)(25,30,39,40)(31,34,42,47)(32,38,44,51)(35,43,49,50)(45,46,48,52);(5,7,10,16)(6,11,13,14)(8,15,9,18)(12,19,17,20)(21,23,27,33)(22,28,26,41)(24,34,37,47)(25,38,39,51)(29,42,36,31)(30,44,40,32)(35,48,49,45)(43,52,50,46);(2,4)(21,24,22,29,27,37,26,36)(23,31,41,47,33,42,28,34)(25,35,30,43,39,49,40,50)(32,45,51,52,44,48,38,46);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(25,39)(30,40)(32,44)(35,49)(38,51)(43,50)(45,48)(46,52);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(21,25,26,40,27,39,22,30)(23,32,28,38,33,44,41,51)(24,35,36,50,37,49,29,43)(31,45,34,46,42,48,47,52)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=hi,d^2=f,e^2=fh,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, 6719086);