Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884841 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #38035 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1582 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,6)(4,5)(7,8)(17,18,24,37,23,29,22,28)(19,25,38,47,33,42,32,41)(20,34,35,26,21,27,36,43)(30,44,45,39,31,40,46,48);(2,5)(3,7)(9,10)(11,14)(12,13)(15,16)(17,19,23,33)(18,25,29,42)(20,30,21,31)(22,32,24,38)(26,39,43,48)(27,40,34,44)(28,41,37,47)(35,45,36,46);(1,3,4,7)(2,6,5,8)(10,13)(11,15)(17,20,23,21)(18,26,29,43)(19,30,33,31)(22,35,24,36)(25,39,42,48)(27,37,34,28)(32,45,38,46)(40,47,44,41);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,21,23,20)(18,27,29,34)(19,31,33,30)(22,35,24,36)(25,40,42,44)(26,37,43,28)(32,45,38,46)(39,47,48,41);(10,13)(14,16)(17,22,23,24)(18,28,29,37)(19,32,33,38)(20,35,21,36)(25,41,42,47)(26,27,43,34)(30,45,31,46)(39,40,48,44)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=fi,d^2=fhi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6844036);