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 | |
2262 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)(9,10,14,11)(12,17,19,15)(13,16,22,18)(20,24,21,23)(25,26)(27,28)(29,37)(30,31)(32,34)(33,40)(35,39)(36,38);(5,6,7,8)(10,11)(12,19)(16,18)(20,21)(26,31)(27,34)(33,37)(35,38);(5,6)(7,8)(9,11,14,10)(12,17,19,15)(13,16,22,18)(20,23,21,24)(25,27)(26,32)(28,31)(29,38)(30,34)(33,36)(35,40)(37,39);(1,2)(3,4)(9,12)(10,15)(11,17)(13,21)(14,19)(16,24)(18,23)(20,22)(25,28,30,32)(26,27,31,34)(29,39,40,36)(33,38,37,35);(9,13)(10,16)(11,18)(12,20)(14,22)(15,23)(17,24)(19,21)(25,29)(26,33)(27,35)(28,36)(30,40)(31,37)(32,39)(34,38)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,c^2=fg,d^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, 3242384);