Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
2322 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)(11,15)(12,14)(13,16)(17,18)(19,23)(20,22)(21,24)(25,26,30,33)(27,34,31,28)(29,37,38,32)(35,36,39,40);(5,6,7,8)(10,12)(11,17)(16,20)(19,21)(27,31)(28,34)(35,39)(36,40);(5,6)(7,8)(9,11,14,17)(10,15,12,18)(13,21,22,19)(16,24,20,23)(25,27,30,31)(26,28,33,34)(29,35,38,39)(32,36,37,40);(1,2)(3,4)(9,12,14,10)(11,18,17,15)(13,16,22,20)(19,23,21,24)(25,28)(26,31)(27,33)(29,36)(30,34)(32,39)(35,37)(38,40);(9,13,14,22)(10,16,12,20)(11,19,17,21)(15,23,18,24)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40)Polycyclic presentation:
2^9|a^2=g,b^2=fgh,c^2=fg,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3242215);