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×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1680 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)(21,22,27,26)(23,28,33,41)(24,29,37,36)(25,30,40,39)(31,34,42,48)(32,38,44,46)(35,43,51,50)(45,47,49,52);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,23,27,33)(22,28,26,41)(24,34,37,48)(25,38,40,46)(29,42,36,31)(30,44,39,32)(35,49,51,45)(43,52,50,47);(2,4)(21,24,22,29,27,37,26,36)(23,31,41,48,33,42,28,34)(25,35,30,43,40,51,39,50)(32,45,46,52,44,49,38,47);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(21,22,27,26)(23,28,33,41)(24,29,37,36)(25,39,40,30)(31,34,42,48)(32,46,44,38)(35,50,51,43)(45,52,49,47);(7,15)(10,13)(14,20)(18,19)(21,25,22,30,27,40,26,39)(23,32,41,46,33,44,28,38)(24,35,29,43,37,51,36,50)(31,45,48,52,42,49,34,47)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=hi,d^2=fi,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, 6716010);