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×#10253 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #120 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1378 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)(5,13)(6,10)(7,16)(8,12)(9,14)(11,17)(15,18)(19,20,24,28)(21,29,31,34)(22,27,23,26)(25,32,33,30);(3,5,8,6)(4,9,12,18)(7,15,17,14)(10,16,13,11)(19,21,24,31)(20,25,28,33)(22,32,23,30)(26,34,27,29);(3,6,8,5)(4,10,12,13)(7,14,17,15)(9,11,18,16)(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, 144761);