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 | #42992 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1637 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)(3,7)(4,6)(5,8)(9,10)(11,16)(12,14)(13,15)(17,18,24,30)(19,25,33,42)(20,26,31,40)(21,27,23,29)(22,36,39,28)(32,47,48,41)(34,45,37,43)(35,46,38,44);(1,3)(2,5)(4,8)(6,7)(17,19,24,33)(18,25,30,42)(20,21,31,23)(22,37,39,34)(26,27,40,29)(28,45,36,43)(32,35,48,38)(41,44,47,46);(9,11,13,12)(10,14,15,16)(17,20,23,19,24,31,21,33)(18,26,29,25,30,40,27,42)(22,37,38,48,39,34,35,32)(28,45,46,47,36,43,44,41);(2,6)(5,7)(17,21,24,23)(18,27,30,29)(19,31,33,20)(22,35,39,38)(25,40,42,26)(28,44,36,46)(32,37,48,34)(41,45,47,43);(3,8)(5,7)(9,12,13,11)(10,14,15,16)(17,22,21,35,24,39,23,38)(18,28,27,44,30,36,29,46)(19,32,20,34,33,48,31,37)(25,41,26,43,42,47,40,45)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gh,d^2=i,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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, 6741836);