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 | #52294 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1550 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(9,10)(11,15)(12,13)(14,16)(17,18,23,27,22,26,31,21)(19,20,25,30,28,24,32,29);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,26)(21,30)(22,28)(23,25)(27,29)(31,32);(10,13)(14,15)(17,20,23,30,22,24,31,29)(18,25,27,28,26,32,21,19);(2,5)(6,8)(9,11,12,16)(10,14,13,15)(19,28)(20,24)(25,32)(29,30);(1,3,4,7)(2,6,5,8)(17,21,31,26,22,27,23,18)(19,29,32,24,28,30,25,20)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=efi,c^b=cgi,d^b=df,e^b=ei,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6338815);