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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2224 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)(9,10,14,12)(11,15,17,18)(13,16,22,20)(19,23,21,24)(25,26,30,32)(27,33,34,28)(29,37,38,31)(35,36,40,39);(5,6,7,8)(10,12)(11,17)(16,20)(19,21)(27,34)(28,33)(35,40)(36,39);(5,6)(7,8)(9,11)(10,15)(12,18)(13,21)(14,17)(16,24)(19,22)(20,23)(25,27)(26,28)(29,35)(30,34)(31,36)(32,33)(37,39)(38,40);(1,2)(3,4)(9,12,14,10)(11,18,17,15)(13,16,22,20)(19,23,21,24)(25,28)(26,27)(29,36)(30,33)(31,35)(32,34)(37,40)(38,39);(9,13,14,22)(10,16,12,20)(11,19,17,21)(15,23,18,24)(25,29,30,38)(26,31,32,37)(27,35,34,40)(28,36,33,39)Polycyclic presentation:
2^9|a^2=fg,b^2=fgh,d^2=f,e^2=fg,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, 3242596);