Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #332327 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
833 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,18)(12,15)(13,17)(14,21)(16,22)(19,24)(20,23);(9,11)(10,14)(12,20)(13,19)(15,18)(16,24)(17,23)(21,22);(2,5)(3,8)(6,7)(11,19)(20,21);(2,5)(3,8)(6,7)(10,15)(11,20)(12,22)(16,17)(18,24)(19,21)Polycyclic presentation:
2^9|e^2=h,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=de,e^b=eh,f^b=fh,f^c=fi,e^d=eh,f^d=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72238923, #72238934, #72238935 |
| maximal normal subgroup of | #72238923, #72238934, #72238935 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 239523);