Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #107918 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22⋊#95 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
872 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,7,5,3,8,6)(9,10,14,16)(11,13,18,22)(12,15)(17,19)(20,24)(21,23);(1,3)(2,5)(4,6)(7,8)(9,11,13,16,14,18,22,10)(12,17,21,24,20,23,19,15);(1,2)(3,5)(4,7)(6,8)(9,12)(10,15)(11,17)(13,21)(14,20)(16,24)(18,23)(19,22)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=h,e^2=i,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,d^b=di,e^b=eh,d^c=dh,e^c=ei,f^c=fhi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72238984, #72239111, #72239112, … |
| maximal normal subgroup of | #72238984, #72239111, #72239112, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 15131);