Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2471 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)(25,26,30,28)(27,32,35,31)(29,33,38,36)(34,40,37,39);(6,8)(10,12)(11,18)(13,21)(22,23)(25,27,30,35)(26,31,28,32)(29,37,38,34)(33,40,36,39);(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)(29,34)(30,35)(33,40)(36,39)(37,38);(1,2)(3,4)(9,12,14,10)(11,15,18,17)(13,20,21,16)(19,22,24,23)(25,28,30,26)(27,32,35,31)(29,36,38,33)(34,40,37,39);(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,29)(26,33)(27,34)(28,36)(30,38)(31,39)(32,40)(35,37)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=g,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1446830);