Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2286 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(6,8)(13,14)(15,20)(16,18)(17,19);(5,6,7,8)(9,10,12,11)(14,18)(19,20);(14,19)(15,17)(18,20);(9,11,12,10)(13,15,16,17)(14,20,18,19)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=eh,f^2=i,b^a=be,c^a=cf,f^a=fi,d^c=di,f^c=fi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72216479, C4×#21503, #72264283, … |
| maximal normal subgroup of | #72216479, C4×#21503, #72264283, … |
| maximal quotient of | #72216458, #72242479 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 262139);