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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2272 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)(11,16,19,23)(12,20,22,13)(17,18,21,24)(25,26,31,34)(27,32,28,29)(30,38,39,33)(35,37,36,40);(6,8)(10,15)(13,20)(16,23)(18,24)(25,27,31,28)(26,29,34,32)(30,36,39,35)(33,40,38,37);(5,6)(7,8)(9,11)(10,16)(12,21)(13,18)(14,19)(15,23)(17,22)(20,24)(25,28,31,27)(26,32,34,29)(30,36,39,35)(33,40,38,37);(1,2,3,4)(9,12)(10,13)(11,17)(14,22)(15,20)(16,24)(18,23)(19,21)(25,29,31,32)(26,28,34,27)(30,37,39,40)(33,36,38,35);(9,13)(10,12)(11,18)(14,20)(15,22)(16,21)(17,23)(19,24)(25,30,31,39)(26,33,34,38)(27,35,28,36)(29,37,32,40)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=g,d^2=fgi,e^2=g,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, 1455277);