Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #106 |
| commutator | C8 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×#340 |
| prime radical | C24 | #103 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×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:
(1,2,4,7,5,8,3,6)(9,10,12,18,16,21,15,31)(11,22,25,37,27,38,23,34)(13,20,14,30,28,36,29,19)(17,26,33,24,35,40,32,39);(2,6,8,7)(9,11,14,26,12,23,28,39,16,27,29,40,15,25,13,24)(10,17,20,34,31,33,19,22,21,35,36,37,18,32,30,38);(9,12,16,15)(10,18,21,31)(11,23,27,25)(13,14,28,29)(17,32,35,33)(19,20,30,36)(22,34,38,37)(24,26,39,40)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=i,d^2=g,e^2=h,f^2=gh,g^2=i,b^a=bd,d^a=dg,f^a=fg,g^a=gi,e^b=ei128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33971);