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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2302 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)(11,17,18,15)(13,20,21,16)(19,22,24,23)(25,26)(27,28)(29,37)(30,31)(32,33)(34,38)(35,40)(36,39);(5,6,7,8)(11,18)(13,21)(15,17)(16,20)(26,31)(27,33)(29,38)(36,40);(5,6)(7,8)(9,11,14,18)(10,15,12,17)(13,19,21,24)(16,22,20,23)(25,27)(26,32)(28,31)(29,39)(30,33)(34,36)(35,38)(37,40);(1,2)(3,4)(9,12,14,10)(11,15,18,17)(13,20,21,16)(19,22,24,23)(25,28,30,32)(26,33,31,27)(29,40,38,36)(34,35,37,39);(9,13,14,21)(10,16,12,20)(11,19,18,24)(15,22,17,23)(25,29,30,38)(26,34,31,37)(27,35,33,39)(28,36,32,40)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,c^2=fg,d^2=g,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, 4040990);