Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438017 |
| 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 | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1900 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,8)(4,6)(5,7)(9,10,12,14)(11,13,15,16)(17,18)(19,26)(20,24)(21,31)(22,25)(23,29)(27,32)(28,30);(9,11)(10,13)(12,15)(14,16)(17,19,21,29,22,27,20,30)(18,23,24,26,25,28,31,32);(1,3,5,4)(2,6,7,8)(11,15)(13,16)(19,27)(23,28)(26,32)(29,30);(1,4,5,3)(2,6,7,8)(11,15)(13,16)(17,18)(19,28)(20,31)(21,24)(22,25)(23,27)(26,30)(29,32);(17,20,22,21)(18,24,25,31)(19,29,27,30)(23,32,28,26)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=g,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6366076);