Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10374 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #393 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1439 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
767 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(9,10,13,17,14,18,12,16)(11,19,22,15,21,23,20,24);(1,2)(3,5)(4,7)(6,8)(11,20,21,22)(15,23,24,19);(2,5)(3,8)(6,7)(9,11)(10,15)(12,22)(13,20)(14,21)(16,23)(17,19)(18,24)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=hi,f^2=hi,g^2=h,b^a=bd,c^a=ce,e^a=ei,f^a=fhi,c^b=cf,e^b=ei,f^b=fi,d^c=dh,e^c=ehi,f^c=fhi,g^c=gh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 23495);