Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| automorphism | ? |
| 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 | |
1812 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,4)(5,7)(10,14)(12,16)(17,18)(19,22)(20,24)(21,23)(25,26,30,31)(27,28,32,36)(29,33,37,38)(34,35,39,40);(1,2,6,8)(3,7,4,5)(17,19,21,24)(18,22,23,20)(26,31)(27,32)(29,37)(35,40);(1,3,6,4)(2,7,8,5)(9,10,13,14)(11,12,15,16)(17,19)(18,20)(21,24)(22,23)(25,27)(26,28)(29,34)(30,32)(31,36)(33,35)(37,39)(38,40);(1,4,6,3)(2,5,8,7)(9,11,13,15)(10,12,14,16)(25,28)(26,32)(27,31)(29,35)(30,36)(33,39)(34,38)(37,40);(1,5)(2,3)(4,8)(6,7)(9,12)(10,15)(11,14)(13,16)(17,20)(18,19)(21,22)(23,24)(25,29)(26,33)(27,34)(28,35)(30,37)(31,38)(32,39)(36,40)Polycyclic presentation:
2^9|a^2=f,b^2=fhi,c^2=fgh,d^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1028096);