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 | #36684 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1467 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,28,22,27,30,38)(19,24,21,26,31,39,36,43)(20,32,35,47,34,25,40,42)(29,44,33,46,45,37,48,41);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,29)(21,23)(22,31)(25,37)(26,28)(27,39)(30,36)(32,44)(33,35)(34,45)(38,43)(40,48)(41,42)(46,47);(10,14)(11,15)(17,20,23,35,22,34,30,40)(18,25,28,42,27,32,38,47)(19,29,21,33,31,45,36,48)(24,37,26,41,39,44,43,46);(2,5)(6,8)(10,14)(13,16)(17,21)(18,26)(19,30)(20,33)(22,36)(23,31)(24,38)(25,41)(27,43)(28,39)(29,40)(32,46)(34,48)(35,45)(37,47)(42,44);(1,3,4,7)(2,6,5,8)(19,31)(20,34)(21,36)(24,39)(25,32)(26,43)(35,40)(42,47)Polycyclic presentation:
2^9|a^2=h,c^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cgi,d^b=dfi,e^b=ei,e^c=ei,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6319091);