Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1461 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,5,7,8)(9,10,12,14)(11,13,15,16)(18,22)(19,25)(24,29)(27,31)(33,34)(35,40)(36,42)(37,41)(38,39)(43,45)(44,48)(46,47);(3,7)(5,8)(9,11,12,15)(10,13,14,16)(20,28)(24,29)(27,31)(30,32)(34,39)(35,42)(41,47)(43,48);(1,3,4,7)(2,5,6,8)(9,11)(10,13)(12,15)(14,16)(17,18,21,22)(19,26,25,23)(20,24,28,29)(27,32,31,30)(33,35,38,42)(34,40,39,36)(37,43,46,48)(41,45,47,44);(9,10)(11,13)(12,14)(15,16)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32)(33,36,38,40)(34,35,39,42)(37,45,46,44)(41,48,47,43);(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,37,38,46)(34,41,39,47)(35,43,42,48)(36,44,40,45)Polycyclic presentation:
2^9|a^2=fh,b^2=fgi,c^2=fhi,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5660418);