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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2425 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)(11,17)(12,16)(13,14)(15,18)(19,24)(20,22)(21,23)(25,26,30,32)(27,33,34,28)(29,37,38,31)(35,36,40,39);(5,6,7,8)(10,14)(11,18)(16,20)(19,21)(27,34)(28,33)(35,40)(36,39);(5,6)(7,8)(9,10)(11,15)(12,20)(13,14)(16,22)(17,18)(19,24)(21,23)(25,27)(26,28)(29,35)(30,34)(31,36)(32,33)(37,39)(38,40);(1,2)(3,4)(9,11,13,18)(10,15,14,17)(12,21,22,19)(16,24,20,23)(25,28)(26,27)(29,36)(30,33)(31,35)(32,34)(37,40)(38,39);(9,12)(10,16)(11,19)(13,22)(14,20)(15,23)(17,24)(18,21)(25,29,30,38)(26,31,32,37)(27,35,34,40)(28,36,33,39)Polycyclic presentation:
2^9|a^2=g,b^2=fgh,d^2=f,e^2=g,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, 3242579);