Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632020 |
| 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 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1404 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)(4,7)(5,8)(9,10,12,15,11,13,16,14)(17,18)(19,25)(20,28)(21,24)(22,27)(23,30)(26,31)(29,32);(1,3,6,2)(4,7,5,8)(10,13)(14,15)(17,19,21,27)(18,22,24,25)(20,29,30,26)(23,32,28,31);(1,4,6,5)(2,7,3,8)(10,14)(12,16)(13,15);(1,5,6,4)(2,7,3,8)(10,13)(14,15)(17,20,21,30)(18,23,24,28)(19,26,27,29)(22,31,25,32);(10,13)(14,15)(18,24)(22,25)(23,28)(31,32)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=h,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfhi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6550162);