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) = C26 |
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 | |
1625 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,7)(9,10,12,13)(11,14,16,15)(19,23)(22,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(2,5)(6,8)(9,10,12,13)(11,15,16,14)(18,21)(19,23)(26,30)(31,33)(32,36)(38,39);(10,13)(14,15)(17,18,20,21)(19,22,23,24)(25,26)(27,33)(28,36)(29,30)(31,35)(32,37)(34,38)(39,40);(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,22)(20,23)(21,24)(25,27,29,35)(26,31,30,33)(28,34,37,40)(32,38,36,39);(1,3)(2,6)(4,7)(5,8)(9,11,12,16)(10,14,13,15)(25,28,29,37)(26,32,30,36)(27,34,35,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=fgh,d^2=fhi,e^2=fg,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, 6203928);