Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1369 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)(4,5)(6,8)(11,15)(13,16)(17,18,21,24)(19,22,25,29)(20,27,28,23)(26,32,31,30)(33,34,39,41)(35,37,36,43)(38,40,46,48)(42,45,44,47);(2,5)(3,8)(19,25)(22,29)(26,31)(30,32)(33,35,39,36)(34,37,41,43)(38,44,46,42)(40,47,48,45);(1,2,4,5)(3,6,8,7)(9,10,12,14)(11,13,15,16)(17,19,21,25)(18,22,24,29)(20,26,28,31)(23,30,27,32)(33,36,39,35)(34,37,41,43)(38,44,46,42)(40,45,48,47);(1,3)(2,6)(4,8)(5,7)(9,11,12,15)(10,13,14,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,37)(34,36)(35,41)(38,45)(39,43)(40,44)(42,48)(46,47);(1,2)(3,6)(4,5)(7,8)(17,20,21,28)(18,23,24,27)(19,26,25,31)(22,30,29,32)(33,38,39,46)(34,40,41,48)(35,42,36,44)(37,45,43,47)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=fgi,d^2=hi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5243457);