Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10260 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#10541 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | #121 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 30 points:
(1,2);(3,4)(5,6)(7,8,12,14)(9,10,13,11)(16,20)(17,23)(18,26)(21,29)(22,27)(24,28);(7,9)(8,10)(11,14)(12,13)(15,16,19,22)(17,24,23,21)(18,27,26,20)(25,28,30,29);(4,6)(8,13)(9,14)(10,11)(15,17,18,25,19,23,26,30)(16,21,20,28,22,24,27,29)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 146237);