Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18414 |
| 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 | C24 | C4×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
822 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,9,7,11,5,10)(3,12,8,15,14,16,13,4)(17,18)(19,28)(20,29)(21,25)(22,26)(23,27)(24,30)(31,32);(1,3,6,8,7,14,5,13)(2,4,9,12,11,15,10,16)(17,19)(18,24)(20,27)(21,29)(22,30)(23,31)(25,28)(26,32);(1,4,7,15)(2,8,11,13)(3,9,14,10)(5,16,6,12)(17,20,23,24)(18,19,27,31)(21,28,22,32)(25,30,26,29);(1,5,7,6)(2,9,11,10)(3,13,14,8)(4,12,15,16)(17,21,23,22)(18,25,27,26)(19,29,31,30)(20,32,24,28)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,d^2=h,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fh,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^c=fh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 99777);