Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10500 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#10503 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #394 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(4,9)(5,10)(6,7)(11,12)(13,16)(14,17)(15,19)(18,23)(20,24)(21,26)(22,25);(3,4)(5,6)(7,9)(8,10)(11,13,15,21)(12,16,19,26)(14,22,23,20)(17,25,18,24);(3,5,6,4,8,9,7,10)(12,17)(14,23)(16,24)(18,19)(20,22)(25,26)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 147942);