Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C23 | C4×D8 |
| prime radical | C24 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
548 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,4)(5,6,8,7)(9,10,15,22)(11,24,28,19)(12,30,31,38)(13,32,14,34)(16,23,17,36)(18,27,37,26)(20,25,21,33)(29,40,35,39);(2,4)(9,11,16,29,15,28,17,35)(10,18,23,30,22,37,36,38)(12,25,27,14,31,33,26,13)(19,20,39,32,24,21,40,34);(5,7,8,6)(9,12,15,31)(10,19,22,24)(11,25,28,33)(13,29,14,35)(16,27,17,26)(18,20,37,21)(23,39,36,40)(30,32,38,34)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=hi,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,d^b=dh,e^b=eh,h^b=hi,d^c=dh,e^c=eh,h^c=hi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54445);