Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×D4×#41 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4940 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)(9,10,13,16,12,14,19,21)(11,15,18,22,17,20,23,24);(1,2)(3,4)(10,14)(15,20)(16,21)(22,24);(6,8)(9,11)(10,15)(12,17)(13,18)(14,20)(16,22)(19,23)(21,24);(2,4)(10,14)(15,20)(16,21)(22,24);(11,17)(15,20)(18,23)(22,24)Polycyclic presentation:
2^9|a^2=h,h^2=i,b^a=bf,c^a=cg,d^a=di,d^b=dfi,e^c=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6286192);