Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1237 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,5)(3,6,7,8)(10,14)(11,12)(17,18,24,30)(19,25,33,41)(20,26,38,46)(21,27,23,29)(22,28,32,40)(31,39,47,48)(34,42,36,44)(35,43,37,45);(9,10)(11,16)(12,15)(13,14)(17,19,24,33)(18,25,30,41)(20,34,38,36)(21,32,23,22)(26,42,46,44)(27,40,29,28)(31,35,47,37)(39,43,48,45);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,20,21,35,24,38,23,37)(18,26,27,43,30,46,29,45)(19,31,22,36,33,47,32,34)(25,39,28,44,41,48,40,42);(9,12,13,11)(10,15,14,16)(17,21,24,23)(18,27,30,29)(19,32,33,22)(20,35,38,37)(25,40,41,28)(26,43,46,45)(31,34,47,36)(39,42,48,44);(1,3,4,7)(2,6,5,8)(17,22,23,19,24,32,21,33)(18,28,29,25,30,40,27,41)(20,36,37,31,38,34,35,47)(26,44,45,39,46,42,43,48)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=hi,d^2=fi,e^2=gh,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, 6740687);