Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7493 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
| direct product | C2×#7493 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1390 |
| commutator | C2×D4 | C2×C42 |
| Frattini, cosocle | C2×#127 | C23 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C22×#127 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
886 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2);(3,4)(5,12,14,18)(6,11,16,9)(7,10)(8,13,17,15);(3,5)(4,8,10,18)(6,15)(7,13)(9,17,11,12)(14,16)56 conjugacy classes — view the full character table.
| maximal subgroup of | #72250643, #72250646, #72250648, … |
| maximal normal subgroup of | #72250643, #72250646, #72250648, … |
| maximal quotient of | #72279605, #72284664, #72284706, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59009);