Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| 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 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1293 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)(10,14)(11,12)(17,18,23,28)(19,24,30,38)(20,25,33,41)(21,26,36,44)(22,27,32,40)(29,37,46,48)(31,39,34,42)(35,43,45,47);(9,10)(11,15)(12,16)(13,14)(17,19,23,30)(18,24,28,38)(20,31,33,34)(21,22,36,32)(25,39,41,42)(26,27,44,40)(29,45,46,35)(37,47,48,43);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,20,32,45,23,33,22,35)(18,25,40,47,28,41,27,43)(19,29,36,34,30,46,21,31)(24,37,44,42,38,48,26,39);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(20,33)(25,41)(29,46)(31,34)(35,45)(37,48)(39,42)(43,47);(10,14)(15,16)(17,21,22,30,23,36,32,19)(18,26,27,38,28,44,40,24)(20,34,35,29,33,31,45,46)(25,42,43,37,41,39,47,48)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fhi,d^2=g,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6729805);