Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29730 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,24,21,19,20,23);(1,2)(3,4)(10,13)(14,16)(17,19,20,24,21,18,22,23);(2,4)(9,11)(10,14)(12,15)(13,16)(17,18,21,19)(20,24,22,23);(11,15)(14,16)(17,20,21,22)(18,23,19,24)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=i,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,c^b=cg,d^c=de,f^c=fi,g^c=gi,h^c=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287308);