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) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1530 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)(19,24)(20,26)(21,25)(22,23)(27,32)(28,29)(30,31)(35,41)(37,45)(39,42)(40,44);(2,5)(6,8)(10,13)(14,16)(18,23)(19,26)(25,31)(27,29)(33,34,38,36)(35,42,41,39)(37,44,45,40)(43,46,47,48);(9,10,12,13)(11,14,15,16)(17,19,22,26)(18,24,23,20)(21,29,30,27)(25,28,31,32)(33,35,38,41)(34,39,36,42)(37,43,45,47)(40,46,44,48);(1,2)(3,6)(4,5)(7,8)(17,20,22,24)(18,19,23,26)(21,28,30,32)(25,27,31,29)(33,36,38,34)(35,39,41,42)(37,44,45,40)(43,46,47,48);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,32)(26,29)(33,37,38,45)(34,40,36,44)(35,43,41,47)(39,46,42,48)Polycyclic presentation:
2^9|b^2=g,c^2=h,d^2=fg,e^2=g,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, 6044602);