Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| 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 | |
1865 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(11,15)(13,16)(17,18,21,22)(19,23,24,20)(25,26,30,31)(27,34,33,29)(28,36,37,32)(35,39,40,38);(3,8)(5,7)(9,10,12,14)(11,13,15,16)(19,24)(20,23)(26,31)(27,33)(28,37)(38,39);(1,2)(3,5)(4,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,20)(21,24)(22,23)(25,27,30,33)(26,29,31,34)(28,35,37,40)(32,39,36,38);(1,3,4,8)(2,5,6,7)(9,10)(11,13)(12,14)(15,16)(17,20)(18,19)(21,23)(22,24)(25,28,30,37)(26,32,31,36)(27,35,33,40)(29,39,34,38);(1,2)(3,5)(4,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,20)(21,22)(23,24)(25,29)(26,33)(27,31)(28,38)(30,34)(32,35)(36,40)(37,39)Polycyclic presentation:
2^9|a^2=fhi,b^2=fghi,c^2=f,d^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2565696);