Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3974 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(9,10,11,12)(13,14,17,18)(15,19,21,25)(16,23,24,20)(22,28,27,26);(2,4)(5,6,7,8)(14,18)(15,21)(20,23)(22,27);(1,2,3,4)(5,6)(7,8)(13,14)(15,19)(16,20)(17,18)(21,25)(22,26)(23,24)(27,28);(13,15)(14,19)(16,22)(17,21)(18,25)(20,26)(23,28)(24,27);(13,16,17,24)(14,20,18,23)(15,22,21,27)(19,26,25,28)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=fg,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 422536);