Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C24 |
| direct product | C2×#7193 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #99 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C23 | #324 |
| prime radical | C25 | M16 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
926 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2);(7,8,10,15,12,13,18,20,11,16,19,22,9,14,17,21);(3,4,5,6)(8,13,16,14)(15,21,22,20)104 conjugacy classes — view the full character table.
| maximal subgroup of | #72214477, #72215373, #72216739, … |
| maximal normal subgroup of | #72214477, #72215373, #72216739, … |
| maximal quotient of | C2×#93375 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34119);