Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #490472 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#466 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | #19101 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
487 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,15,19)(11,20,24,37)(12,17,14,29)(13,27,28,40)(16,30,33,23)(18,26,34,21)(22,35,25,36)(31,39,32,38);(2,5)(6,7)(9,11,21,38,14,22,28,30,15,24,26,39,12,25,13,23)(10,16,27,36,17,32,34,37,19,33,40,35,29,31,18,20);(2,6)(4,8)(5,7);(9,12,15,14)(10,17,19,29)(11,22,24,25)(13,26,28,21)(16,31,33,32)(18,27,34,40)(20,35,37,36)(23,38,30,39)Polycyclic presentation:
2^9|a^2=i,b^2=egh,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,g^a=gi,d^b=di,f^c=fh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72309442, #72309456, #72309458, … |
| maximal normal subgroup of | #72309442, #72309456, #72309458, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 397771);