Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10238 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#10249 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1486 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4)(5,9)(6,8)(7,11)(10,14)(12,17)(13,18)(15,16)(19,20,24,28)(21,29,31,34)(22,27,23,26)(25,32,33,30);(4,8)(6,14)(9,17)(10,11)(12,15)(16,18)(19,21)(20,25)(22,32)(23,30)(24,31)(26,34)(27,29)(28,33);(3,5,7,13)(4,9,11,18)(6,15,14,12)(8,16,10,17)(20,26)(22,23)(25,29)(27,28)(30,32)(33,34)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 144760);