Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| 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 | |
1396 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)(9,10,13,12)(11,14,16,15)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,34,38,36)(35,41,42,39)(37,44,45,40)(43,46,48,47);(1,3,4,7)(2,5,6,8)(9,11,13,16)(10,14,12,15)(18,22)(23,25)(24,28)(30,31)(35,42)(37,45)(39,41)(40,44);(1,3)(2,5)(4,7)(6,8)(9,12,13,10)(11,14,16,15)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,30)(31,32)(33,35,38,42)(34,39,36,41)(37,43,45,48)(40,46,44,47);(9,11)(10,15)(12,14)(13,16)(17,19,21,27)(18,23,22,25)(20,26,29,32)(24,30,28,31)(33,36,38,34)(35,39,42,41)(37,44,45,40)(43,46,48,47);(1,2)(3,5)(4,6)(7,8)(17,20,21,29)(18,24,22,28)(19,26,27,32)(23,30,25,31)(33,37,38,45)(34,40,36,44)(35,43,42,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=fh,b^2=fgh,c^2=fgi,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6144041);