Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1608 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)(3,5)(4,6)(7,8)(9,10,12,14)(11,13,15,16)(17,18,21,20)(19,22,23,24)(26,30)(27,33)(28,35)(39,40);(2,3)(6,7)(19,23)(22,24)(25,26,29,30)(27,31,33,38)(28,32,35,36)(34,39,37,40);(4,8)(6,7)(17,19)(18,22)(20,24)(21,23)(25,26)(27,31)(28,36)(29,30)(32,35)(33,38)(34,40)(37,39);(1,3,5,2)(4,6,8,7)(9,10)(11,13)(12,14)(15,16)(25,27)(26,31)(28,37)(29,33)(30,38)(32,40)(34,35)(36,39);(1,4)(2,6)(3,7)(5,8)(9,11,12,15)(10,13,14,16)(17,20,21,18)(19,24,23,22)(25,28)(26,32)(27,34)(29,35)(30,36)(31,39)(33,37)(38,40)Polycyclic presentation:
2^9|a^2=fghi,b^2=g,d^2=fg,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3554146);