Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
2264 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,3,4)(10,14)(11,17)(12,19)(22,24)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,39)(38,40);(6,8)(10,14)(11,17)(16,21)(18,23)(25,27,31,29)(26,32,34,28)(30,35,38,37)(33,39,40,36);(5,6,7,8)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,28,31,32)(26,27,34,29)(30,36,38,39)(33,35,40,37);(1,2)(3,4)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,29,31,27)(26,32,34,28)(30,35,38,37)(33,36,40,39);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,30,31,38)(26,33,34,40)(27,35,29,37)(28,36,32,39)Polycyclic presentation:
2^9|a^2=gi,b^2=f,c^2=h,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2515443);