Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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 | |
1392 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(4,7)(11,15)(13,16)(18,22)(23,25)(24,29)(30,31)(33,34,37,38)(35,41,43,39)(36,40,45,44)(42,47,48,46);(1,2,5,6)(3,7,8,4)(9,10,12,14)(11,13,15,16)(20,28)(24,29)(26,32)(30,31)(34,38)(39,41)(40,44)(46,47);(1,3)(2,7)(4,6)(5,8)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,34)(35,41)(36,44)(37,38)(39,43)(40,45)(42,46)(47,48);(9,11,12,15)(10,13,14,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,35,37,43)(34,39,38,41)(36,42,45,48)(40,46,44,47);(1,4,5,7)(2,8,6,3)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=h,b^2=fi,d^2=ghi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5429477);