Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10192 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#10205 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| 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 | |
1396 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);(3,4)(5,8)(6,10)(7,9)(11,12,17,22)(13,23,14,25)(15,21,16,20)(18,26,19,24);(4,8)(5,7)(9,10)(11,13)(12,18)(14,17)(15,26)(16,24)(19,22)(20,25)(21,23);(3,4,6,10)(5,8,7,9)(11,14,17,13)(12,19,22,18)(15,26,16,24)(20,25,21,23)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 144577);