Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #2682 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1415 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,30,33,29,28,32,37)(27,31,35,38,34,36,39,40);(1,3,4,7)(2,5,6,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27,29,34)(26,31,28,36)(30,35,32,39)(33,38,37,40);(2,6)(3,7)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(26,28)(27,34)(33,37)(35,39);(2,6)(5,8)(11,19)(14,17)(18,24)(21,23)(25,28,30,37,29,26,32,33)(27,31,35,38,34,36,39,40);(10,16)(14,17)(15,22)(21,23)(27,34)(31,36)(35,39)(38,40)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=f,d^2=h,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dfi,e^b=ei,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72308911, #72308965, #72308985, … |
| maximal normal subgroup of | #72308911, #72308965, #72308985, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6329424);