Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #116849 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #388 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1440 |
| prime radical | C24 | #120 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
647 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,6,4,7,3,5,8)(9,10,13,19,14,17,12,18)(11,20,21,24,22,15,23,16);(1,3,7,2)(4,5,8,6)(9,11)(10,15)(12,21)(13,23)(14,22)(16,18)(17,20)(19,24);(1,4,5,2,7,8,6,3)(9,11)(10,16,17,24)(12,21)(13,23)(14,22)(15,18,20,19)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=d,d^2=i,e^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=di,f^b=fi,g^b=ghi,g^c=gh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 26411);