Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19713 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×#127 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
766 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,4,3)(5,6)(7,9)(8,11)(10,12)(15,21)(16,23)(18,24)(20,26)(22,27)(25,28);(5,6,9,11,10,12,8,7)(13,14)(15,18)(16,20)(17,19)(21,24)(22,25)(23,26)(27,28);(5,7,8,12,10,11,9,6)(13,15,16,21,17,22,23,27)(14,18,20,24,19,25,26,28);(1,3,4,2)(5,8,10,9)(6,11,12,7)(14,19)(18,25)(20,26)(24,28)Polycyclic presentation:
2^9|a^2=g,b^2=eh,c^2=fh,d^2=gh,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=di,d^c=dh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 120090);