Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1324 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,5,7,8)(10,13)(14,16)(17,18,22,23)(19,25,27,21)(20,28,29,24)(26,31,30,32)(35,41)(36,44)(38,42)(39,46);(11,15)(14,16)(18,23)(20,29)(21,25)(26,30)(33,34,37,40)(35,42,41,38)(36,39,44,46)(43,48,47,45);(9,10)(11,14)(12,13)(15,16)(17,19)(18,21)(20,30)(22,27)(23,25)(24,32)(26,29)(28,31)(33,35,37,41)(34,38,40,42)(36,43,44,47)(39,45,46,48);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,20,22,29)(18,24,23,28)(19,26,27,30)(21,31,25,32)(33,34)(35,38)(36,39)(37,40)(41,42)(43,45)(44,46)(47,48);(1,3,4,7)(2,5,6,8)(17,21)(18,19)(20,31)(22,25)(23,27)(24,26)(28,30)(29,32)(33,36)(34,39)(35,43)(37,44)(38,45)(40,46)(41,47)(42,48)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,d^2=f,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4458868);