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 | |
503 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)(3,5)(4,6)(7,8)(9,10)(11,19)(12,16)(13,22)(14,24)(15,18)(17,20)(21,23);(2,5)(6,7)(9,11,20,18,14,21,23,24,15,17,19,10,12,22,13,16);(2,6)(4,8)(5,7);(9,12,15,14)(10,16,18,24)(11,21,17,22)(13,19,23,20)Polycyclic presentation:
2^9|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 | #72342429, #72342436, #72342437, … |
| maximal normal subgroup of | #72342429, #72342436, #72342437, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 397770);