Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10518 |
| 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 | C2×#532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1058 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,7)(3,8,6,4)(9,10,14,19)(11,12,16,23)(13,17,24,18)(15,20,22,21);(2,3)(4,8)(6,7)(9,11)(10,15)(12,17)(13,20)(14,16)(18,23)(19,22)(21,24);(1,3,4,2,5,6,8,7)(9,12,13,22,14,23,24,15)(10,16,18,20,19,11,17,21);(9,13,14,24)(10,17,19,18)(11,20,16,21)(12,22,23,15)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=hi,e^2=h,f^2=i,g^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,g^b=gi,f^c=fhi,g^c=ghi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 146069);