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 | #392 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1287 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)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,27,21,26,35,43)(19,23,30,37,29,25,40,42)(20,31,34,45,33,24,38,39)(28,41,44,48,32,36,46,47);(1,3,6,2)(4,7,5,8)(9,11,12,15)(10,13,14,16)(17,19,21,29)(18,23,26,25)(20,28,33,32)(22,30,35,40)(24,36,31,41)(27,37,43,42)(34,44,38,46)(39,47,45,48);(10,14)(11,15)(17,20,22,34,21,33,35,38)(18,24,27,39,26,31,43,45)(19,28,30,44,29,32,40,46)(23,36,37,47,25,41,42,48);(1,4,6,5)(2,7,3,8)(10,14)(13,16)(17,19)(18,25)(20,32)(21,29)(22,30)(23,26)(24,36)(27,42)(28,33)(31,41)(34,46)(35,40)(37,43)(38,44)(39,47)(45,48);(1,5,6,4)(2,7,3,8)(19,29)(20,33)(23,25)(24,31)(30,40)(34,38)(37,42)(39,45)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=h,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dfi,e^b=ei,d^c=di,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, 6320621);