Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806575 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,6,7,8)(9,10,12,14)(11,15,16,13)(17,18,21,24)(19,27,28,23)(20,29,31,25)(22,26,30,32);(9,11)(10,13)(12,16)(14,15)(17,19,21,28)(18,23,24,27)(20,22,31,30)(25,26,29,32);(2,5)(6,8)(10,14)(11,16)(17,19,22,20,21,28,30,31)(18,23,26,25,24,27,32,29);(10,14)(13,15)(18,24)(23,27)(25,29)(26,32);(1,3,4,7)(2,6,5,8)(17,20,30,19,21,31,22,28)(18,25,32,23,24,29,26,27)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=fhi,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6739709);