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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2313 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)(9,10,14,15)(11,12,17,16)(13,21,24,18)(19,23,22,20)(25,26)(27,33)(28,31)(29,32)(30,35)(34,40)(36,38)(37,39);(6,8)(10,15)(11,17)(18,21)(19,22)(25,26,29,32)(27,33,35,30)(28,31,38,36)(34,40,37,39);(5,6)(7,8)(9,11,14,17)(10,16,15,12)(13,22,24,19)(18,20,21,23)(25,26)(27,30)(28,36)(29,32)(31,38)(33,35)(34,40)(37,39);(1,2)(3,4)(9,12,14,16)(10,17,15,11)(13,23,24,20)(18,19,21,22)(25,27,29,35)(26,30,32,33)(28,37,38,34)(31,40,36,39);(9,13,14,24)(10,18,15,21)(11,19,17,22)(12,20,16,23)(25,28,29,38)(26,31,32,36)(27,34,35,37)(30,39,33,40)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=g,d^2=f,e^2=f,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, 3240163);