Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10012 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#10017 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | #392 |
| prime radical | C24 | #120 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1438 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);(3,4,8,11)(5,12,14,18)(6,10,7,15)(9,16,17,13)(19,20)(21,27)(22,30)(23,26)(24,29)(25,32)(28,34)(31,33);(3,5,8,14)(4,9,11,17)(6,13,7,16)(10,18,15,12)(24,33)(27,34);(3,6,8,7)(4,10,11,15)(5,13,14,16)(9,12,17,18)(19,21)(20,24)(22,31)(23,28)(25,34)(26,33)(27,30)(29,32)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 140764);