Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #36687 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1678 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)(11,13)(12,14)(15,16)(17,18,22,26,21,25,35,42)(19,27,30,43,29,23,36,38)(20,31,34,45,33,24,39,41)(28,37,44,47,32,40,46,48);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,37)(25,27)(26,38)(31,40)(32,33)(34,44)(35,36)(39,46)(41,47)(42,43)(45,48);(10,14)(11,15)(17,20,22,34,21,33,35,39)(18,24,26,41,25,31,42,45)(19,28,30,44,29,32,36,46)(23,37,38,47,27,40,43,48);(3,8)(5,7)(10,14)(13,16)(17,19)(18,23)(20,32)(21,29)(22,30)(24,40)(25,27)(26,38)(28,33)(31,37)(34,46)(35,36)(39,44)(41,48)(42,43)(45,47);(2,6)(5,7)(19,29)(23,27)(28,32)(30,36)(37,40)(38,43)(44,46)(47,48)Polycyclic presentation:
2^9|a^2=h,c^2=h,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cgi,d^b=dfi,e^b=ei,d^c=di,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6318521);