Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #612424 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
6866 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6,7,8)(9,10)(11,12)(13,14)(15,16)(18,21)(19,23);(10,14)(11,15)(17,18,20,21)(19,22,23,24);(5,6)(7,8)(9,11,13,15)(10,12,14,16);(1,2)(3,4)(5,6)(7,8);(9,12)(10,15)(11,14)(13,16)(17,19)(18,22)(20,23)(21,24)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch116 conjugacy classes — view the full character table.
| maximal subgroup of | #72329203, #72329217, #72339910, … |
| maximal normal subgroup of | #72329203, #72329217, #72339910, … |
| maximal quotient of | #72329848, #72336133, #72339926, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 520213);