Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #186176 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #62835 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2272 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,8,6,3,4,7)(9,10,14,19,15,17,13,18)(11,20,22,12,16,24,21,23);(1,3)(2,6)(4,8)(5,7)(9,11,15,16)(10,12,17,23)(13,22,14,21)(18,24,19,20);(1,2)(3,6)(4,8)(5,7)(9,12)(10,16)(11,17)(13,20)(14,24)(15,23)(18,22)(19,21);(1,4,6,5)(2,7,3,8)(10,17)(12,23)(18,19)(20,24)Polycyclic presentation:
2^9|a^2=g,b^2=hi,d^2=h,e^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,e^b=ei,g^b=ghi,d^c=di,f^c=fh,g^c=gh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 93564);