Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10144 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1438 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
870 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5,6,7)(9,10,13,15,12,16,20,22)(11,14,19,23,18,24,21,17);(1,2)(3,7)(4,6)(5,8)(9,11,12,18)(10,14,16,24)(13,21,20,19)(15,17,22,23);(1,2)(3,7)(4,6)(5,8)(9,10)(11,17)(12,16)(13,22)(14,21)(15,20)(18,23)(19,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=i,e^2=hi,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=di,e^b=ehi,f^b=fh,g^b=ghi,d^c=dhi,e^c=ehi,f^c=fh,g^c=ghi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 21395);