Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| 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 | |
1582 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)(10,13)(14,15)(17,18,21,24)(19,22,23,20)(25,26,29,30)(27,33,35,31)(28,36,37,32)(34,38,40,39);(2,5)(6,8)(9,10,12,13)(11,15,16,14)(19,23)(20,22)(26,30)(31,33)(32,36)(38,39);(10,13)(14,15)(17,19,21,23)(18,22,24,20)(25,27,29,35)(26,31,30,33)(28,34,37,40)(32,38,36,39);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,20,21,22)(18,23,24,19)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,16)(13,15)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40)Polycyclic presentation:
2^9|a^2=fh,b^2=fg,c^2=fh,d^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5876753);