Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10326 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| 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 | |
725 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)(9,10,13,16,14,17,12,21)(11,18,19,15,20,24,22,23);(1,2)(3,7)(4,6)(5,8)(11,19,20,22)(15,18,23,24);(2,6)(3,4)(7,8)(9,11)(10,15)(12,22)(13,19)(14,20)(16,24)(17,23)(18,21)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=i,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=ei,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 23455);