Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10456 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #394 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#394 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
962 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,7)(3,5,8,6)(9,10)(11,19)(12,21)(13,17)(14,23)(15,18)(16,20)(22,24);(9,11)(10,16)(12,17)(13,20)(14,24)(15,21)(18,22)(19,23);(2,5)(3,8)(6,7)(9,12,14,16,15,19,13,22)(10,11,17,20,18,21,23,24);(9,13,15,14)(10,17,18,23)(11,20,21,24)(12,22,19,16)Polycyclic presentation:
2^9|a^2=i,c^2=e,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gh,c^b=cg,e^b=eh,f^b=fh,f^c=fhi,g^c=gh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 147252);