Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #36694 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(9,10)(11,15)(12,13)(14,16)(17,18,21,25,20,22,26,29)(19,23,28,30,27,24,31,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(18,22)(23,24)(25,29)(30,32);(10,13)(14,15)(17,19,21,28,20,27,26,31)(18,23,25,30,22,24,29,32);(2,5)(6,8)(9,11,12,16)(10,14,13,15);(1,3,4,7)(2,6,5,8)(17,19,26,31,20,27,21,28)(18,24,29,30,22,23,25,32)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=g,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dfi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6338820);