Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18252 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#385 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
865 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10,7,12,5,11)(3,13,14,4,15,16,9,8)(17,18,22,26)(19,27,30,31)(20,23,21,25)(24,29,32,28);(1,3,5,9,7,15,6,14)(2,8,11,16,12,4,10,13)(18,23)(20,21)(24,31)(25,26)(27,32)(28,29);(1,4)(2,9)(3,10)(5,13)(6,16)(7,8)(11,15)(12,14)(17,19)(18,24)(20,28)(21,29)(22,30)(23,31)(25,27)(26,32);(1,5,7,6)(2,10,12,11)(3,14,15,9)(4,13,8,16)(17,20,22,21)(18,25,26,23)(19,28,30,29)(24,27,32,31)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=h,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fh,d^b=dh,e^b=eh,f^b=fi,e^c=ei,f^c=fh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 99740);