Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ #1744 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
832 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,8,11,9,7,10,12)(14,18)(16,23)(19,24)(20,25)(22,27)(26,28);(13,14,16,18,17,20,23,25)(15,19,22,24,21,26,27,28);(1,2,3,4)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28);(5,7,8,12,9,6,10,11)(15,21)(19,26)(22,27)(24,28)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=f,d^2=g,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,d^c=dh128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 120123);