Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | |
2291 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)(10,14)(11,17)(12,20)(22,24)(25,26)(27,34)(28,29)(30,31)(32,37)(33,39)(35,40)(36,38);(5,6,7,8)(12,20)(16,21)(19,23)(22,24)(26,31)(29,39)(32,34)(36,40);(5,6)(7,8)(9,10,13,14)(11,18,17,15)(12,16,20,21)(19,24,23,22)(25,27)(26,32)(28,38)(29,40)(30,37)(31,34)(33,35)(36,39);(1,2)(3,4)(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24)(25,28)(26,29)(27,35)(30,33)(31,39)(32,36)(34,40)(37,38);(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,29)(26,33)(27,36)(28,31)(30,39)(32,38)(34,35)(37,40)Polycyclic presentation:
2^9|b^2=h,c^2=fg,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,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, 4040917);