Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10046 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#10061 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | #120 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1466 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(7,8,12,15)(9,16,18,21)(10,14,11,19)(13,20,22,17)(23,24,28,32)(25,33,36,37)(26,29,27,31)(30,34,38,35);(3,4)(5,6)(7,9,12,18)(8,13,15,22)(10,17,11,20)(14,21,19,16)(24,29)(26,27)(30,37)(31,32)(33,38)(34,35);(4,6)(7,10,12,11)(8,14,15,19)(9,17,18,20)(13,21,22,16)(23,25,28,36)(24,30,32,38)(26,35,27,34)(29,37,31,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 142860);