Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #616960 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2416 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3340 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(6,7)(9,10,12,13)(11,15,16,14)(17,18)(19,21)(20,23)(22,24);(10,13)(14,15)(18,19)(22,23);(5,6,8,7)(17,18,21,19)(20,22,24,23);(1,2)(3,4)(5,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,19,21,18)(20,23,24,22);(9,11,12,16)(10,14,13,15)(17,20,21,24)(18,22,19,23)Polycyclic presentation:
2^9|a^2=fh,c^2=g,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 524454);