Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10004 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#10011 |
| 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 | |
1342 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);(3,4,9,14)(5,15,6,17)(7,13,8,12)(10,18,11,16)(19,20,24,28)(21,29,22,26)(23,32,33,27)(25,31,34,30);(3,5,9,6)(4,10,14,11)(7,18,8,16)(12,17,13,15)(19,21,24,22)(20,25,28,34)(23,31,33,30)(26,32,29,27);(3,6,9,5)(4,11,14,10)(7,18,8,16)(12,17,13,15)(19,22,24,21)(20,26,28,29)(23,30,33,31)(25,27,34,32)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 142518);