Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
2226 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,4)(9,10,14,17)(11,12,18,15)(13,21,22,16)(19,23,24,20)(25,26,31,33)(27,34,28,30)(29,37,39,32)(35,38,36,40);(6,8)(11,18)(12,15)(19,24)(20,23)(25,27,31,28)(26,30,33,34)(29,35,39,36)(32,38,37,40);(5,6,7,8)(9,11)(10,15)(12,17)(13,19)(14,18)(16,23)(20,21)(22,24)(25,28,31,27)(26,30,33,34)(29,35,39,36)(32,40,37,38);(1,2,3,4)(9,12,14,15)(10,11,17,18)(13,20,22,23)(16,19,21,24)(25,29,31,39)(26,32,33,37)(27,35,28,36)(30,38,34,40);(9,13,14,22)(10,16,17,21)(11,19,18,24)(12,20,15,23)(25,30)(26,27)(28,33)(29,38)(31,34)(32,35)(36,37)(39,40)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=fgh,d^2=i,e^2=g,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, 1521163);