Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #763332 |
| 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 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,4)(9,10,13,14)(11,15,16,12)(19,23)(22,24);(5,6,7,8)(10,14)(11,16)(18,21)(22,24);(1,2)(3,4)(5,6)(7,8)(10,14)(12,15)(17,18)(19,22)(20,21)(23,24);(9,11,13,16)(10,15,14,12)(17,19)(18,22)(20,23)(21,24);(9,12,13,15)(10,16,14,11)Polycyclic presentation:
2^9|a^2=g,b^2=hi,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df,e^c=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 673138);