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 | C2×#469 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1679 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)(21,22,26,30)(23,27,32,38)(24,28,35,41)(25,29,36,42)(31,37,43,47)(33,39,45,49)(34,40,46,50)(44,48,51,52);(5,7,10,16)(6,11,13,14)(8,15,9,18)(12,19,17,20)(23,31)(24,34)(25,36)(27,37)(28,40)(29,42)(32,43)(33,45)(35,46)(38,47)(39,49)(41,50);(2,4)(21,23,25,31,26,32,36,43)(22,27,29,37,30,38,42,47)(24,33,34,44,35,45,46,51)(28,39,40,48,41,49,50,52);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(23,32)(27,38)(31,43)(33,45)(37,47)(39,49)(44,51)(48,52);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(21,24,25,34,26,35,36,46)(22,28,29,40,30,41,42,50)(23,33,31,44,32,45,43,51)(27,39,37,48,38,49,47,52)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=h,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,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6718910);