Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10128 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#10142 |
| 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 | |
1252 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,8,5)(6,13,14,18)(7,12,17,10)(9,16,11,15)(19,20)(21,28)(22,25)(23,27)(24,30)(26,32)(29,34)(31,33);(3,5,8,4)(6,11,14,9)(7,10,17,12)(13,16,18,15)(19,21,23,31)(20,24,27,34)(22,29,32,30)(25,28,26,33);(3,6)(4,9)(5,11)(7,16)(8,14)(10,18)(12,13)(15,17)(20,25)(21,29)(22,32)(26,27)(28,33)(30,31)62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 143919);