Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| automorphism | Aut(G) order 2097152 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1776 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,7)(3,4,5,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,20)(19,24,21,23)(25,26,29,30)(27,31,34,36)(28,32,35,37)(33,38,39,40);(1,3,6,5)(2,4,7,8)(9,11)(10,13)(12,16)(14,15)(26,30)(28,35)(31,36)(33,39);(2,7)(4,8)(10,12)(11,15)(17,19,22,21)(18,23,20,24)(25,26)(27,31)(28,32)(29,30)(33,38)(34,36)(35,37)(39,40);(1,4,6,8)(2,3,7,5)(9,12,14,10)(11,16,15,13)(17,20,22,18)(19,23,21,24)(25,27,29,34)(26,31,30,36)(28,33,35,39)(32,38,37,40);(1,5,6,3)(2,8,7,4)(9,13,14,16)(10,15,12,11)(17,21,22,19)(18,23,20,24)(25,28)(26,32)(27,33)(29,35)(30,37)(31,38)(34,39)(36,40)Polycyclic presentation:
2^9|a^2=fgh,b^2=fhi,c^2=fgi,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4883237);