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×#466 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1672 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)(23,34)(24,37)(25,40)(26,32)(27,33)(28,31)(29,36)(30,39)(35,50)(38,48)(41,52)(42,46)(43,49)(44,47)(45,51);(5,7,10,16)(6,11,13,14)(8,15,9,18)(12,19,17,20)(21,23)(22,29)(24,25)(26,42)(27,36)(28,39)(30,40)(31,37)(32,48)(33,34)(35,41)(38,43)(44,51)(45,52)(46,49)(47,50);(2,4)(21,24,25,23,27,39,28,36)(22,30,40,29,33,37,31,34)(26,38,41,35,43,42,44,51)(32,46,52,45,49,48,47,50);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(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);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(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=fi,b^2=fi,c^2=hi,d^2=f,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, 6718922);