Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32888 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3900 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,14,18,15,17,13,16)(11,19,20,12,22,23,21,24);(1,2,3,4)(9,11)(10,12)(13,20)(14,21)(15,22)(16,23)(17,24)(18,19);(9,12)(10,11)(13,23)(14,19)(15,24)(16,20)(17,22)(18,21);(6,8)(9,13,15,14)(10,16,17,18)(11,20,22,21)(12,23,24,19)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,e^b=ei,f^b=fi,h^b=hi,e^c=ei,h^c=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299845);