Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #380391 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2331 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14286 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
615 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,12,5,10)(3,13,15,25,16,24,22,8)(4,17,20,27,21,9,19,28)(14,29,30,23,31,26,18,32);(1,3,5,15,7,16,6,22)(2,8,11,24,12,25,10,13)(4,18,20,31,21,30,19,14)(9,26,27,32,17,29,28,23);(1,4,7,21)(2,9)(3,14,16,31)(5,19,6,20)(8,23)(10,27)(11,28)(12,17)(13,26)(15,30,22,18)(24,29)(25,32);(4,19,21,20)(9,27,17,28)(14,30,31,18)(23,26,32,29)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=eg,d^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 288279);