Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4311730 |
| 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 | |
1362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,3)(6,7)(10,13)(14,16)(17,18,22,21)(19,25,27,23)(20,28,30,24)(26,31,29,32)(36,43)(39,46)(41,47)(44,48);(18,21)(23,25)(24,28)(31,32)(33,34,37,40)(35,38,42,45)(36,39,43,46)(41,44,47,48);(1,2,5,3)(4,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,29)(21,25)(22,27)(24,32)(26,30)(28,31)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(1,3,5,2)(4,7,8,6)(17,20)(18,24)(19,26)(21,28)(22,30)(23,31)(25,32)(27,29)(33,35,37,42)(34,38,40,45)(36,41,43,47)(39,44,46,48);(1,4,5,8)(2,6,3,7)(9,11,12,15)(10,14,13,16)(17,21,22,18)(19,25,27,23)(20,28,30,24)(26,32,29,31)(33,36,37,43)(34,39,40,46)(35,41,42,47)(38,44,45,48)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=fi,d^2=fhi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4264739);