Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #238361 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#393 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
909 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,17,20,15)(14,16)(18,22)(19,23,24,21);(3,4)(6,7)(9,11,13,10)(12,19)(14,18,22,16)(15,21)(17,23)(20,24);(2,6)(3,4)(7,8)(9,12)(10,15)(11,17)(13,20)(14,23)(16,19)(18,24)(21,22);(1,3,5,4)(2,7,8,6)(12,20)(15,17)(19,24)(21,23)Polycyclic presentation:
2^9|a^2=eh,b^2=e,d^2=hi,f^2=hi,g^2=h,b^a=be,c^a=cf,f^a=fi,g^a=ghi,c^b=cg,d^b=dhi,f^b=fi,d^c=di,e^c=eh,f^c=fhi,g^c=gh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 145561);