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×#10198 |
| 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 | |
1384 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);(3,4,8,12)(5,13,15,18)(6,11,7,10)(9,16,17,14)(19,20,24,28)(21,29,32,33)(22,27,23,26)(25,31,34,30);(3,5,8,15)(4,9,12,17)(6,16,7,14)(10,18,11,13)(19,21,24,32)(20,25,28,34)(22,31,23,30)(26,33,27,29);(3,5)(4,9)(6,14)(7,16)(8,15)(10,13)(11,18)(12,17)(21,30)(22,23)(25,29)(26,27)(31,32)(33,34)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 144566);