Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #321983 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2541 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #19701 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
757 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,7,4,6,8,5)(9,10,14,19,15,20,13,16)(11,12,21,23,18,22,24,17);(10,16,20,19)(11,21,18,24)(12,22)(17,23);(2,5)(3,8)(6,7)(9,11)(10,17)(12,16)(13,21)(14,24)(15,18)(19,22)(20,23);(2,6)(5,7)(9,12,13,17,15,22,14,23)(10,18,19,24,20,11,16,21)Polycyclic presentation:
2^9|a^2=f,b^2=eg,d^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,d^b=dh,f^c=fi,g^c=gh50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 229179);