Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#8408 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#538 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
3414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(3,4)(5,6)(7,8)(9,10,14,12)(11,13)(15,16,18,21,19,17,22,20);(4,6)(7,9)(8,12)(10,13)(11,14)(15,17)(16,19)(18,20)(21,22);(7,10,11,12)(8,9,13,14)(18,22)(20,21)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 93122);