Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667484 |
| 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 | C22×D8 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#125 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1396 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)(3,6)(4,5)(7,8)(17,18,21,24)(19,25,27,22)(20,23,28,30)(26,31,32,29);(9,10,13,16)(11,15,12,14)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31);(2,5)(6,8)(10,14)(11,12)(15,16)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31);(1,3,4,7)(2,6,5,8)(18,24)(22,25)(23,30)(29,31);(9,11)(12,13)(14,15)(19,27)(22,25)(26,32)(29,31)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=f,d^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6605608);