Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10378 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #393 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1439 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
723 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,6)(4,7)(5,8)(9,10,13,17,14,16,12,22)(11,18,21,24,20,15,19,23);(1,2)(3,7)(4,6)(5,8)(11,19,20,21)(15,23,18,24);(2,6)(3,4)(7,8)(9,11)(10,15)(12,21)(13,19)(14,20)(16,18)(17,24)(22,23)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=hi,f^2=i,g^2=h,b^a=bd,c^a=ce,e^a=ei,f^a=fh,c^b=cf,f^b=fhi,d^c=dh,e^c=ehi,f^c=fi,g^c=gh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 23468);