Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| 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 | |
1425 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)(11,15)(14,16)(17,18,22,21)(19,23,25,27)(20,28,29,24)(26,32,30,31)(33,34)(35,37)(36,43)(38,46)(39,41)(40,47)(42,48)(44,45);(2,5)(6,8)(10,13)(14,16)(18,21)(19,25)(20,29)(31,32)(33,35,39,36)(34,37,41,43)(38,42,47,44)(40,45,46,48);(9,10,12,13)(11,14,15,16)(17,19)(18,23)(20,30)(21,27)(22,25)(24,32)(26,29)(28,31)(33,36,39,35)(34,37,41,43)(38,44,47,42)(40,45,46,48);(1,2)(3,6)(4,5)(7,8)(17,20,22,29)(18,24,21,28)(19,26,25,30)(23,31,27,32)(33,37)(34,36)(35,41)(38,45)(39,43)(40,44)(42,46)(47,48);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,21,22,18)(19,27,25,23)(20,28,29,24)(26,32,30,31)(33,38,39,47)(34,40,41,46)(35,42,36,44)(37,45,43,48)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fgh,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5868320);