Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1516 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,13,15)(11,12,16,14)(17,18,21,23)(19,24,22,20)(26,30)(27,33)(28,35)(38,40);(9,11,13,16)(10,14,15,12)(17,19,21,22)(18,20,23,24)(26,30)(27,33)(32,37)(34,39);(10,15)(12,14)(17,20,21,24)(18,22,23,19)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,2)(3,5)(4,6)(7,8)(9,12,13,14)(10,11,15,16)(17,20)(18,19)(21,24)(22,23)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(1,3,4,7)(2,5,6,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,20)(21,22)(23,24)(25,28)(26,32)(27,34)(29,35)(30,37)(31,38)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=fgh,d^2=fh,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1814102);