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 | |
1480 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)(17,18,24,20,23,27,21,31)(19,25,29,28,30,22,26,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,25)(20,28)(21,26)(22,27)(23,30)(24,29)(31,32);(10,13)(14,15)(17,20,24,27,23,31,21,18)(19,28,29,22,30,32,26,25);(2,5)(6,8)(9,11,12,16)(10,14,13,15)(17,21,23,24)(18,20,27,31)(19,29,30,26)(22,28,25,32);(1,3,4,7)(2,6,5,8)(17,22,24,32,23,25,21,28)(18,26,20,19,27,29,31,30)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=cgi,d^b=dfi,d^c=di,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6338826);