Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6343554 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| prime radical | #62835 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #3444 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2785 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,6)(5,8)(9,12)(10,14)(11,13)(15,16)(17,18,22,28,23,26,21,27)(19,29,31,42,32,24,36,38)(20,25,33,40,34,39,35,41)(30,43,45,47,46,37,44,48);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(17,19)(18,24)(20,30)(21,31)(22,36)(23,32)(25,37)(26,29)(27,38)(28,42)(33,44)(34,46)(35,45)(39,43)(40,47)(41,48);(1,3,5,10)(2,7,8,14)(4,11,12,16)(6,13,9,15)(17,20,23,34)(18,25,26,39)(19,30,32,46)(21,35,22,33)(24,37,29,43)(27,41,28,40)(31,45,36,44)(38,48,42,47);(2,8)(3,10)(6,9)(11,16)(18,26)(24,29)(25,39)(27,28)(37,43)(38,42)(40,41)(47,48);(3,10)(7,14)(11,16)(13,15)(17,21,23,22)(18,27,26,28)(19,31,32,36)(20,33,34,35)(24,38,29,42)(25,40,39,41)(30,44,46,45)(37,47,43,48)Polycyclic presentation:
2^9|a^2=g,c^2=h,e^2=hi,f^2=i,g^2=hi,b^a=bf,d^a=dh,f^a=fh,f^b=fi,g^b=ghi,d^c=di,e^c=eh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251438);