Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10088 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#10101 |
| 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 | |
1286 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)(6,9)(7,11)(8,12)(10,14)(15,17)(16,18)(19,20,24,28)(21,29,22,26)(23,32,33,27)(25,31,34,30);(4,9)(6,7)(10,13)(11,12)(14,16)(17,18)(19,21,24,22)(20,25,28,34)(23,31,33,30)(26,32,29,27);(3,5,8,15)(4,10,12,18)(6,14,7,16)(9,17,11,13)(19,22,24,21)(20,26,28,29)(23,30,33,31)(25,27,34,32)62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 143452);