Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | |
2178 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)(11,12,18,16)(13,21,22,17)(19,24,23,20)(25,26)(27,30)(28,36)(29,33)(31,35)(32,38)(34,40)(37,39);(6,8)(10,15)(11,18)(13,22)(20,24)(25,27,29,35)(26,30,33,31)(28,37,38,34)(32,40,36,39);(5,6)(7,8)(9,11,14,18)(10,16,15,12)(13,23,22,19)(17,20,21,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,40)(36,39)(37,38);(1,2)(3,4)(9,12,14,16)(10,11,15,18)(13,24,22,20)(17,23,21,19)(25,26)(27,30)(28,32)(29,33)(31,35)(34,39)(36,38)(37,40);(9,13,14,22)(10,17,15,21)(11,19,18,23)(12,20,16,24)(25,28,29,38)(26,32,33,36)(27,34,35,37)(30,39,31,40)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=f,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,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, 4038776);