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 | |
1412 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)(9,10,12,14)(11,13,15,16)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,31)(30,32)(33,34,37,41)(35,39,43,38)(36,40,45,48)(42,47,44,46);(2,5)(6,8)(11,15)(13,16)(18,22)(19,26)(20,29)(31,32)(33,35,37,43)(34,38,41,39)(36,42,45,44)(40,46,48,47);(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,30)(21,26)(22,25)(24,32)(27,29)(28,31)(33,35)(34,39)(36,44)(37,43)(38,41)(40,46)(42,45)(47,48);(1,2)(3,6)(4,5)(7,8)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,29)(30,32)(33,34)(35,38)(36,40)(37,41)(39,43)(42,46)(44,47)(45,48);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,24)(19,27)(21,29)(22,28)(23,31)(25,32)(26,30)(33,36,37,45)(34,40,41,48)(35,42,43,44)(38,46,39,47)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=fh,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, 6045383);