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×#10100 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1404 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)(19,20,23,26)(21,27,29,34)(22,30,32,25)(24,31,33,28);(4,9)(6,7)(10,13)(11,12)(14,16)(17,18)(19,21,23,29)(20,24,26,33)(22,31,32,28)(25,34,30,27);(3,5)(4,10)(6,14)(7,16)(8,15)(9,17)(11,13)(12,18)(22,32)(24,33)(25,30)(28,31)62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 143450);